Download Algebra 1 unit 5

Algebra 1 Unit 5
Solving Systems of Linear Equations
Content to be learned

Solve systems of linear equations by graphing, substitution, and linear combinations.

Show that a system of equations could have one solution, no solution, or infinite solutions.

Connect real-world problem scenarios to linear mathematical representations

Use an effective method (graphing, substitution, elimination) to solve a system of linear equations
formed by a problem scenario.

Interpret the solution to a system of linear equations in a context problem situation

Graph single linear inequality in two variables

Graph systems of linear inequalities on the coordinate plane

Represent constraints as linear inequalities in order to solve real-world situations with linear programming.
Systems of Linear Equations
You already know how to:


Graph lines using slope and y-intercept (when the equation is in slope intercept
2
form such as y =   3 ) and x-and y-intercept (when the equation is in
3
standard form such as 3x – 2y = 6).
Give the coordinates (x, y) of a point that is graphed.
Warm-up:
1) Give the coordinates of each point.
A(
)
B(
)
C(
)
D(
)
E(
)
F(
)
G(
)
2) Graph each line.
a) y = 
2
x 1
3
b) 2x – 3y = 6
A system of linear equations is a fancy way to say “one or more equations”. We will use
3 different methods to solve systems involving only 2 equations. To solve a system
means to tell where the graphs would cross if we graphed them, or which (x, y)
point makes each equation true.
Examples: Determine if the point is a solution to the system of equations.
IXL alg 1 U1
The methods we will use are:



Graphing
Substitution
Elimination
The Graphing Method: To solve a system by graphing, use the slope and y-intercept,
or x-and y-intercepts to graph the two lines accurately on graph paper using a ruler and
give the point (x, y) where they cross. Try these examples. Remember to use a ruler.
 y  2x  1
1) 
 y  x  4
 y  2 x

2) 
1
 y   2 x  3
4 x  y  2
3) 
 x  y  3
2 x  5 y  10
4) 
3x  y  2
Practice
Suppose you graphed these two lines.
Where do they cross?
Are you sure??
The Substitution Method
The method of solving "by substitution" works by plugging one of the equations (the one
that says y = or x =) into the other equation, "substituting" for the chosen variable and
solving for the other. Then you solve for the variable. Here is how it works.
 y  4 x  24
Example 1: Solve the following system by substitution. 
2 x  3 y  2
Plug this in ("substitute it") for "y" in the second equation, and solve for x:
2x – 3(–4x + 24) = –2
Now I can plug this x-value back into either equation, and solve for y. But since I already
have an expression for "y =", it will be simplest to just plug into the first equation:
y = –4(5) + 24 = –20 + 24 = 4
Then the solution is (x, y) = (5, 4).
 y  5 x  13

Example 2: Solve using substitution: 3x  3 y  15
Substitute the first equation in for y in the second equation:
Now substitute this x -value into the first equation to find y :
Thus, the solution to the system is (______,_______).
What happens if one of the equations is not solved for x or y? You will have to solve
one of the equations for x or y- which ever looks easier. Try these examples:
 y  3x  5

x  y  3
5 x  7 y  19

 x  y  3
What happens if both equations are solved for y?
 y  4x  2

y  7x  5
Student Practice
The Elimination Method
This method of solving equations involves adding two equations together in order to
eliminate one of the variables. When using this method,
1. Be sure that both equations are written in standard form Ax + By = C
2. You may have to multiply one of the equations by some number in order to
eliminate a variable. If you do this, remember to multiply everything by that
number.
3. Remember to check your answers.
Examples: Solve each system of equations using the elimination method.
 2 x  3 y  14

1) 2 x  2 y  4
 6 x  10 y  26

3)  5 x  5 y  5
4 x  5 y  5

2) 5 x  5 y  5
6 x  4 y  24

4) 3 x  9 y  12
Sometimes, we have to multiply both equations by some number in order to get the variables to
cancel.
Example:
2 x  4 y  4

3x  10 y  6
7 x  5 y  12

 3x  3 y  0
How do you know which method (graphing, substitution, or elimination) to use???
Method
Graphing
Substitution
Elimination
Best used when
you…
…you can easily
graph the
equations and you
don’t need an
accurate answer
…have an equation
already solved for
one variable (e.g.
y = 14x + 2)
When both
equations are in
standard form
Ax + By = C and A
and B are
numbers other
than 1.
Advantages…
Often easier to see
number and quality
of intersections on
a graph. With a
graphing calculator,
it can be the fastest
method since you
don’t have to do
any computation.
Works on all
systems. Reduces
the system to one
variable, making it
easier to solve.
Works on all
systems. Makes it
possible to
combine
equations to
eliminate one
variable.
Comment…
Can lead to
imprecise answers
with non-integer
solutions.
You may have to do
extra work to get
one of the equations
into that form.
Often more
algebraic
manipulation
(multiplication) is
needed to prepare
the equations for
addition.
Graph the 2 lines and tell where they cross.
Graph the 2 lines and tell where they cross.
Graph the 2 lines and tell where they cross.
Solving Systems of Equations ~ Special Cases

1 solution

The line cross

1 ordered pair

The solution is (x, y)




No Solution
Parallel Lines
The lines never cross
Algebraically you would
get 0 = #, which is not
true!!!

Infinitely many solutions
 Same line
 Algebraically you would get
something that looks like
5 = 5 or -3 = -3
Examples: Solve each system of equations.
7 x  y  6

1)  70 x  10 y  10
 y  6x  6

2) 6 x  y  6
Directions: Solve each system of equations. Each solution is either “many solutions” or
“no solution”.
Word Problems
1. A student bought 3 boxes of pencils and 2 boxes of pens for $6. He then
bought 2 boxes of pencils and 4 boxes of pens for $8. Find the cost of each
box of pencils and each box of pens
2. The sum of two numbers is 19, and their difference is 55. What are the two numbers?
3. For the fundraiser, Will sold 225 candy bars. He earns $1 for each almond candy bar
he sells and $0.75 for each caramel candy bar he sells. If he earned a total of $187.50,
how many of each type of candy bar did he sell for the fundraiser?
4. John paid $34 for two algebra and three geometry books. He paid $36 for three
algebra and two geometry books. What is the cost of each book?
5.A hotel is offering the two weekend specials described below.
Plan 1: 3 nights and 4 meals for $233
Plan 2: 3 nights and 3 meals for $226.50.
a. What is a system of equations for the situation?
b. Solve the system. What is the cost per night and the cost per meal?
6. The perimeter of a rectangle is 60 cm. The length is four times the
width. What are the length and the width of the rectangle?
7. A TV station executive is planning the new lineup for next season's shows. On 7.
Monday nights, there will be 5 sitcoms and 6 dramas, for a total of 318 minutes of
programming, not counting commercials. On Tuesday nights, he has scheduled 5
sitcoms and 1 drama, for a total of 153 minutes of non-commercial programming. How
long is each type of show?
8. A landscaping company placed two orders with a nursery. The first order was for 13
bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees,
and totaled $232. The bills do not list the per-item price. What were the costs of one
bush and of one tree?
9.Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher
minus one small pitcher constitutes 2 cups of water. How many cups of water can
each pitcher hold?
10.A test has twenty questions worth 100 points. The test consists of True/False
questions worth 3 points each and multiple choice questions worth 11 points each. How
many multiple choice questions are on the test?
Systems of Linear Inequalities
DO ixl alg 1 T5 with the class
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_in
equalities_precalc/v/graphical-system-of-inequalities
Systems of Linear Inequalities
Application ~ Linear Programming
1. You can work a total of no more than 41 hours each week at your two jobs.
Housecleaning pays $5 per hour and your sales job pays $8 per hour. You need to
earn at least $254 each week to pay your bills.
a. Write a system of inequalities that shows the various numbers of hours you can
work at each job.
b. Graph the solution set below.
2. Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only
allowed to work 13 hours per week. She wants to make at least $65.
a. Write and graph a system of inequalities to represent this situation.
3. Marsha is buying plants and soil for her garden. The soil cost $4 per bag, and the
plants cost $10 each. She wants to buy at least 5 plants and can spend no more than
$100. Write and graph a system of linear inequalities to model the situation.
4. John is packing books into boxes. Each box can hold either 15 small books or 8
large books. He needs to pack at least 35 boxes and at least 350 books. Write
and graph a system of linear inequalities to model the situation.
5. During a family trip, you share the driving with your dad. At most, you are allowed to
drive for three hours. While driving, your maximum speed is 55 miles per hour.
a) Write a system of inequalities describing the possible numbers of hours t and
distance d you may have to drive.
b) Is it possible for you to have driven 160 miles?
Using a Graphing Calculator to Find the Solution to a System of Equations
a. Set up the viewing window. Press window and enter these
numbers: -5, 5, 1, -10, 10, 1, 1
b. Enter the first equation into y1
c. Enter the second equation into y2
d. Press graph
e. Press 2nd then trace then press 5
f. Using the left and right arrow keys (NOT THE UP/DOWN
keys!!) move the blinking cursor to the left of the point of
intersection and press enter
g. Use the right arrow to move the blinking cursor to the right of
the point of intersection. Press enter twice!!