Download Document

7.1 Solving Systems of Equations Using Graph-and-check
Objective: You will be able to solve a system of 2 equations graphically.
Warm-up:
Determine whether (3, 9) is a solution for the equations below:
y = 2x + 3
y = -3x + 18
What about (1, 5)?
What is a system of linear equations?
*
* Example: y = 2x + 3 and y = -3x + 18
Determine if (-1, -2) is a solution of the system below:
3x + 2y = 4
-x + 3y = -5
What about (2, -1)?
There are 3 methods we will learn to solve a system of equations:
1) Graphing
2) Substitution
3) Linear Combinations
STEPS FOR GRAPHING:
1)
both equations in the same coordinate plane. For ease of graphing, you may want
to write each equation in
.
2) Estimate the coordinates of the
.
3) _________ the coordinates algebraically by substituting into each equation of the
original linear system.
What is the solution of a system of linear equations?
Use the graphs to solve each system.
Use a graph to solve each of the following systems. Then check your solution algebraically.
1) 3x + y = 9
x - y = -1
y = ______________
y = ______________
Solution:________
Check the solution:
2) 2y + 4x = 12
2x - y = -10
y = ______________
y = ______________
Solution:________
Check the solution:
3) 4x + 2y = 6
3x - 3y = 9
y = ______________
y = ______________
Solution:________
Check the solution:
7.1 Homework p. 430, 3-10, 12-14, 18-20. Use graphs on the next page for graphs of 12-14
and 18-20.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.1 Solving Systems of Equations with a Graphing Calculator
Objective: You will be able to solve a system of 2 equations using a graphing calculator.
Warm-up:
Write the equation so that y is a function of x.
5y + 25 = 3x
We can find the intersection of two lines on a graphing calculator, which is the goal of
solving a system of equations graphically.
Steps:
1. Get both equations into slope-intercept form
y = mx +b
2. Type both equations into Y1 and Y2, respectively.
3. Select 2nd - TRACE - 5: Intersect
4. Hit Enter 3 times.
Examples:
1.) 2x + 5y = 7
-x + 2y = -8
2.) -x + 2y = 3
2x + y = 4
Story Problems:
The Rosebud Flower Shop has a basic delivery charge of $5 plus a rate of $.25 per mile.
The Beautiful Bouquets Shop has a basic delivery charge of $7 plus a rate of $.20 per mile.
Write a system of equations, and use it to determine the number of miles a delivery must be
for the charges to be equal.
Your school is selling football tickets for a home game. The school sold 35 tickets for $86
on the first day of the sale. Student tickets cost $2 each and non-student tickets cost $3
each. Write a system of equations, and then use it to find the number of student tickets
and the number of non-student tickets sold.
7.2 Solving Linear Systems Using Substitution
Objective: You will be able to solve systems of linear equations by substitution.
Warm-up:
Solve both equations.
6a - 3 + 2a = 13
4(n + 2) - n = 11
Steps to solving linear systems with substitution:
1) ________ one equation for x or y. Choose the easiest one!
(Should have a coefficient of ______!)
2) _______________ the expression from step 1 into the OTHER
equation and solve for the variable that remains.
3) Substitute the answer from step 2 into one of the original
equations to find the __________________ variable.
Use substitution to solve the linear systems:
Example 1:
1.) y = 2x - 3
x + 3y = 5
Example 2:
2.) 2x + 5y = 5
x - 4y = 9
Story Problem:
A fitness club offers two aerobics classes. There are currently 28 people going to the
afternoon class and attendance is increasing at a rate of 2 people per month. There are
currently 16 people going to the night class and attendance is increasing at a rate of 4
people per month. Write a system of equations, and predict when the number of people in
each class will be the same.
7.2 Homework p. 439, 3-18 all & 32.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.2 homework continued…
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.2 Solving Linear Systems Using Substitution
Objective: You will be able to solve systems of linear equations with coefficients that may
not be 1 or -1 with substitution.
Warm-up:
Tell which equation you would choose to solve for x or y in.
WHY?
3x + y = -7
3x - 5y = 4
-2x + 4y = 0
-5x - y = 12
Solve the systems using substitution.
1.) -5x - y = 12
3x - 5y = 4
OPTION 1:
OPTION 2:
2.) 4x - 2y = -4
7x - 5y = -19
Story Problem:
You burned 8 calories per minute on a treadmill and 10 calories per minute on an elliptical
trainer for a total of 560 calories and 60 minutes. How many minutes did you spend on each
machine? Write a system of linear equations and solve.
An adult ticket to a school play costs $5 and a student ticket costs $3. A total of $460
was collected from the sale of 120 tickets. How many student tickets were purchased?
Solve the problem using a linear system of equations.
7.2 day 2 homework, continued.
19. Last year you mowed grass and shoveled snow for 12 households. You earned $225 for
mowing a household’s lawn for the entire year and you earned $200 for shoveling a
household’s walk and driveway for an entire year. You earned a total of $2600 last year.
a. Let x be the number of households you mowed for and let y be the number of
households you shoveled for. Write an equation in x and y that shows the total
number of households you worked for. Then write an equation in x and y that shows
the total amount of money you earned.
b. How many households did you mow the lawn for and how many households did
you shovel the walk and driveway for?
20. Your math teacher tells you that next week’s test is worth 100 points and contains 38
problems. Each problem is worth either 5 points or 2 points. Because you are studying
systems of linear equations, your teacher says that for extra credit you can figure out how
many problems of each value are on the test. How many of each value are there?
21. You exercised on a treadmill for 1.5 hours. You ran at 4 miles per hour, then you
sprinted at 6 miles per hour. If the treadmill monitor says that you ran and sprinted 7
miles, how long did you run at each speed?
Algebra Review 7.1 – 7.2
Solve each of the following by graphing:
1) y = -x + 3
2) y = 2x – 4
y=x+1
y = -1/2 x + 1
Solution: _______
4) 3x + y = 5
-x + y = -7
_______
Solution: _______
Solution: _______
5) x – 2y = 0
3x – y = 0
Solution:
Solution: _______
Solve each of the following by substitution:
6) 3x + y = 3
7) 2x + y = 4
7x + 2y = 1
-x + y = 1
9) 4x + 3y = 31
y = 2x + 7
3) x – y = 1
5x – 4y = 0
8) x + 2y = 1
5x + 3y = -23
10) –12x + y = 15
3x + 2y = 3
Determine if the point given is a solution for the system of equations.
11) –3x + y = 6
12) y = 3x + 6
-x + y = -2
2x – 3y = 10
(-4, -6)
(5 , 0)
7.3 Solve Linear Systems by Adding or Subtracting
Objective: You will be able to solve a system of linear equations using elimination.
WARM-UP:
What is the opposite of 3?
1.
2.
3.
4.
5.
Simplify -5x + 5x
How to Solve Systems Using Elimination:
Make sure both equations are in _______________form
If needed, change the signs of one equation so one variable
has _________________ coefficients.
Add equations to _______________ one variable.
Solve new equation for remaining variable
Substitute value into original equation to find the value of the variable
that was eliminated.
Solve each linear system by eliminating a variable.
1.) 3x + 4y = 8
2.) -3x - 2y = 31
-3x + 5y = 10
5x + 2y = -49
3.) 5x + 6y = 4
7x + 6y = 8
4.) 5x + 4y = 6
x + 4y = 14
Sometimes we have to rearrange the equations first, then solve the system of equations
using addition or subtraction.
5.) 10y - 3x = -41
6.) 4x - 3y = 39
3x - 5y = 16
7y = 4x - 79
7.3 Homework p. 447, 2-8 all, 10-22 even
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.3 homework, continued.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.4 Solve Linear Systems by Multiplying First
Objective: You will be able to solve a system of linear equations by multiplying first and
then using elimination.
7 and 3
Warm-up:
Find the least common multiple of each pair of numbers:
4 and 12
Sometimes we may have a system of equations with variables that do not have opposite
coefficients that can be added in order to eliminate a variable. In these situations, we can
multiply the entire equation, or possibly even both equations, by a number to find the least
common multiple.
For example...
5x + 2y = 16
Multiply by 2
10x + 4y = 32
3x - 4y = 20
+ 3x - 4y = 20
Now we can add them together!
SOLVING A LINEAR SYSTEM USING ELIMINATION
STEP 1: ____________________ one or both of your equations by a constant value so
that the coefficients of x or y are _____________________.
STEP 2: Add the equations so that you ___________________ one of the two variables.
STEP 3: _____________________ for the remaining variable.
STEP 4: Substitute the known value in for the known variable and then solve for the
unknown variable.
STEP 5: Check your solution.
***How do you decide which variable you want to eliminate?
Solve the given linear systems by elimination:
1.) 3x - 3y = 21
2.) 2x + y = -9
8x + 6y = -14
4x + 11y = 9
Sometimes you need to multiply both equations by a constant before you can eliminate a
variable.
3.) 5x + 2y = -18
4.) 2x + 3y = 17
-3x + 7y = 19
3x + 5y = 27
STORY PROBLEMS:
Mr. Alvarado bought a total of 20 pounds of grass seed at the nursery for $168. He paid
$9 per pound for Kentucky bluegrass and $6 per pound for Tall Fescue. Write a system of
equations that can be used to find the amount x (in pounds) of Kentucky bluegrass and the
amount y (in pounds) of Tall Fescue Mr. Alvarado purchased. Then find the solution.
A sports equipment store is having a sale on soccer balls. A soccer coach purchases 10
soccer balls and 2 soccer bags for $155. Another soccer coach purchases 12 soccer balls
and 3 soccer bags for $189. Find the cost of a soccer ball and the cost of a soccer ball bag.
7.4 Homework, page 454, 2- 18 even, 37 & 38
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.4, homework continued.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Review Worksheet on 7.3-7.4
7.5 Special Types of Linear Systems
Objective: You will be able to identify the number of solutions of a linear system.
Warm-up
Determine how many solutions the equation below has without solving.
-2(x - 3) = 6 - 2x
A linear system of equations can have one solution, no solution, or infinitely many solutions.
What would the graph look like if there was one solution?
What would the graph look like if there are no solutions?
What would the graph look like if there are infinitely many solutions?
Tell how many solutions each system has using the graphing method:
1. 3y – 3x = 12
2. 5x + 5y = -30
y= x – 4
3x + 3y = -18
Tell how many solutions each system has using the substitution method:
3. y - 3x = 5
4. 10x + 5y = -15
x=y-5
2x + y = - 3
Tell how many solutions each system has using the elimination method:
5. 4x - 3y = -2
6. -3x + 5y = 6
6x + 4y = 41
6x - 10y = -12
Summary: How can you tell if a system has one, zero, or many solutions if you solve the system using
substitution or elimination?
ONE:
______________________________________________________________________________________
______________________________________________________________________________________
NONE:
______________________________________________________________________________________
______________________________________________________________________________________
MANY:
______________________________________________________________________________________
______________________________________________________________________________________
7.5 Homework page 462, 3 – 10 all, 16-28 even
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7.5, homework continued.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Chapter 7 Test Review
1. Tell whether the ordered pair is
a solution of the linear system.
a. 3x + y = 1
-5x + y = -31
(4, -11)
Circle one:
YES or NO
b. 2x + 5y = 7
-x + 2y = -8
(1, 1)
Circle one:
YES or NO
2. Solve the linear system using
graphing.
y = 3x + 4
y = -2x - 1
Solution: _________
3. Solve the linear system using
graphing.
2x + 7y = 14
5x + 7y = -7
4. When solving a system of
equations, what is it you are
actually trying to find?
5. True or False: If there is a
system of 2 equations in two
variables that has the same slope
and the same y-intercepts, then
there are no solutions.
6. Solve the linear system by using
substitution.
15x + y = 70
3x – 2y = -8
Solution: ________
7. To solve the linear system
below using substitution, plug
which of the following values in for
y in the second equation:
2x + y = -15
-x + 4y = -2
a) -2x
c) -1/2y + 7.5
b) -2x - 15
d) 4y + 2
8. Solve the linear system using
elimination.
11x – 20y = 28
3x + 4y = 36
Solution: ________
Solution: ________
9. Solve the linear system using
elimination.
6x – 5y = 9
9x – 7y = 15
11. Tickets for admission to a high
school football game cost $3 for
students and $5 for adults. During
one game, $2995 was collected
from the sale of 729 tickets.
a. Write a linear system that
represents this situation.
Equation 1:
Equation 2:
10. You pay $24.50 for 10 gallons
and 1 quart of oil at a gas station.
Your friend pays $22 for 8 gallons
of the same gasoline and 2 quarts
of the same oil.
a. Write a linear system that
represents this situation.
Equation 1:
Equation 2:
b. Solve the linear system to find
the cost of one gallon of gas and
one quart of oil. Make sure to
label your answers.
b. How many tickets were sold
to students and how many were
sold to adults?
In problems 12 - 14, tell whether
each linear system has one
solution, no solutions, or infinitely
many solutions.
12. 9x - 15y = 24
6x – 10y = 16
Number of solutions: __________
13. y = -6x - 2
12x + 2y = -6
Number of solutions: _________
c. How much would it cost for 6
gallons of gasoline and 3 quarts of
oil?
14. 4x + 3y = 27
4x – 3y = -27
Number of solutions: __________