Download Unit 7 - Solving Systems of Equations

Unit 7 - Solving Systems of Equations
System of equations: two or more equations with the same set of variables.
Example: y = 4x and y = 4x + 2 together are a system of equations.
The solution for a system of equation is the point in which the two lines intersect.
The solution is written as an ordered pair (x, y)
For each problem determine if the ordered pair is the solution to the system of equations. Your answer
will be YES or NO. SHOW ALL YOUR WORK to justify your answer.
Example 1
Determine if the point (2, 7) is the solution to the following system of equations.
Example 2
Determine if the point (3, -1) is the solution of the system of equations?
y=x–4
y = –3x
Example 3
Determine if the point (-15, -25) is the solution of the system of equations?
y = x – 10
y = 2x + 5
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Practice:
2) Is the point (2, 3) a solution to the system
1) Is the point (4, 12) a solution to the system
y = 3x
y = 2x + 7
5x + 2y = 44
4) Is the point (5, 2) a solution to the system
3) Is the point (3, 4) a solution to the system
x + 3y = 15
3x + y = 17
-8x + 3y = -12
3x + 2y = 11
36 – 4y = 3x
7x – y = 3
8) Is the point (5, -2) a solution to the system
7) Is the point (2, -1) a solution to the system
-4x – 9y = 1
4x – y = 18
6) Is the point (3, 1) a solution to the system
5) Is the point (4, 6) a solution to the system
4y + 4 = 7x
3x – y = -9
x – 5y = 15
-x + 2y = -4
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4x – 3y = 26
Lesson 7.1 - Solving Systems of Equations by graphing.
a)
b)
c)
d)
e)
Label the ____________ and _______________________ for each equation.
_______________ each equation on the same coordinate plane
Find the point of _________________________of the two lines
Write this point as an ___________________. This is the _______________ to the system
_______________ your solution by substituting the coordinates into each equations
Solve the system of equations by graphing each equation
1.
2. y = x – 1
3. y = 3x + 8
y = 2x - 2
y=x–2
3
4.
y = 2x – 4
1
y= 𝑥+3
4
5.
3
y=− 𝑥+4
2
2
6. y = 𝑥 + 4
3
3
y= 𝑥−2
2
y = -2x - 4
4
7. y = x − 4
8. x – 4y = 4
9.
y = -3x + 10
y = −2x + 5
5x – 4y = 12
4y + 3x = 4
5
10. 5y = 2x – 10
y = -4x – 2
11. x – y = 4
x + 2y = -2
12. 4x – 5y = -25
-9x – 2y = 10
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Lesson 7.2 - Solving Systems of Equations by Substitution. DAY 1 (simple substitution)
1. If needed, solve one of the equations for y (_______________________________ form)
2. Replace (__________________________) the y in one equation with the value of y for the
3.
4.
5.
6.
other equation.
Solve for x.
Solve for y (if needed)
Write answer as an ordered pair (x, y). This is the solution to the system
Check your solution by substituting the coordinates into each equations
Solve the system of equations by substitution.
1.
2.
3. y = 7x – 20
y = -6
4. y = 9x
4.
y = -9x -15
5. y = 4x + 25
y = -6x
6. 2x + y = -14
2y = 10x
7. 3y = 9x
7
y = 5x + 12
y = 9x
6x = 4y + 6
Lesson 7.2 - Solving Systems of Equations by Substitution. DAY 2 (distributive property)
1.
2.
3.
4.
5.
6.
1.
If needed, solve one of the equations for y (slope intercept form)
Put parentheses around the expression you are substituting in to the other equation
Replace this expression with the value of y for the other equation.
Solve for x.
Write answer as an ordered pair (x, y). This is the solution to the system
Check your solution by substituting the coordinates into each equations
y = x + 12
4x + 2y = 8
3. y = 3x +8
8x + 4y = 12
5. y = 2x – 3
x + y = 18
7. x + y = 1
5x – 4y = -7
2. 10x + 3y = 19
y = 2x + 5
4. 4x + 2y = 27
y = x + 12
6. y = 2x + 1
-2x + 3y = -9
8. -5x + y = 35
x – 2y = -21
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7.3 - Solving Linear Systems by Linear Combinations (Day 1)
STEPS
1)
Arrange the equations with like terms in columns
2)
IF NEEDED - Multiply one or both of the equations by a number to get
opposite numbers for one of the variables.
3)
Add the equations together.
(Combining like terms will wipe out one variable. Solve for the variable left
over)
4)
Plug in the value from step 3 into either of the original equations and solve
for the other variable.
5)
Check the solution in BOTH of the original equations to make sure it is
true.
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Example 1)
4x + 3y = 16
2x – 3y = 8
Example 2)
x–y=8
x + y = 20
Example 3)
x + 4y = 23
-x + y = 2
Example 4)
2y = x – 8
x + 6y = -16
Example 5)
Remember: 1) Columns
2) Opposite Coefficients
3) Add and Solve
4) Solve other variable
5) Check
y=x-9
x + 8y = 0
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PRACTICE
Use elimination to solve each system of equations:
4)
-6x + 5y = 1
6x + 4y = -10
7)
x – y = 11
2x + y = 19
5)
8)
2x – 3y = 12
4x + 3y = 24
8x = 9 – 5y
-8x + 3y = 31
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6)
2x + y = 9
3x – y = 16
9) 3x + 2y = 0
9x – 24 = 2y
7.4 - Solving by Linear Equations (Day 2)
Today, when we arrange the variables in columns, there will not be opposite
coefficients for one of the variables. Instead, we will have to do some multiplying to
force opposites to develop.
STEPS
1) Arrange the equations with like terms in columns
2) IF NEEDED - Multiply one or both of the equations by a number to get
opposite numbers for one of the variables.
3) Add the equations together.
-Combining like terms will wipe out one variable.
-Solve for the variable left over
4) Plug in the value from step 3 into either of the original equations and solve for
the other variable.
5) Check the solution in BOTH of the original equations to make sure it is true.
Example 1: Solve the system
x – y = -5
x + 2y = 4
Example 2: Solve the system
5x + y = 9
10x – 7y = -18
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Example 3: Solve the system
-7x + y = -19
-2x + 3y = -19
Example 4: Solve the system
16x – 10y = 10
-8x – 6y = 6
Example 5: Solve the system
8x + 14y = 4
-6x – 7y = 10
Example 6: Solve the system
5x + 4y = 9
4x + 5y = 9
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Example 7: Solve the system
5x + 4y = -30
-3x – 9y = -18
Example 8: Solve the system
3x = -6y + 12
1x + 3y = 6
Example 9: Solve the system
3x + 2y = 8
2y = 12 – 5x
Example 10: Solve the system
-2y = - 3x + 2
-5x – 5y = 10
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PRACTICE
Use elimination to solve each system of equations:
1)
3x + 2y = 0
x – 5y = 17
2)
2x + 3y = 6
x + 2y = 6
3)
3x – y = 2
x +2y = 3
4)
4x + 2y = 8
16x – y = 14
5)
4x + 5y = 6
6x – 7y = - 20
6)
10x + 3y = 19
y – 2x = 5
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7.5 - Special Types of Linear Systems
So far we have looked at systems of equations that have exactly one solution(x, y).
But there are two other possible solutions.
One Solution
(Lines Intersect)
NO Solution
(Parallel Lines)
MANY Solutions
(Lines are the same)
GRAPHING
Graph the following systems. Determine who many solutions the system has. If it has one
solution, give the coordinate.
1. y – 2x = 4
y=2x
2. y = 3x – 4
1
y=− 𝑥+3
2
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3. y = -2x + 4
y = -2x - 3
ALGEBRAICALLY (using substitution or elimination
IF BOTH variables are eliminated….
When you get a TRUE statement : 3 = 3, or 0 = 0, or 12 = 12, etc… there are MANY
solutions.
When you get a FALSE statement: 0 = 4, 5 = 1, -3 = 10, etc… there is NO solution.
Example 1
Pick the method of your choice.
2x + y = 5
2x + y = 1
Example 2: Pick the method of your choice.
-x + 2y = -2
3x – 6y = 6
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Example 3: Pick the method of your choice.
-y = 3x + 4
-6x – 2y = -8
Example 4: Pick the method of your choice.
3x + y = -5
6x + 2y = 10
Example 5: Pick the method of your choice.
5x – y = 5
-x + 3y = 13
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Practice:
Solve by graphing
y = -3x + 4
-3y = 9x – 12
y = 5x – 2
-5y = -25x - 15
1.
2.
Solution ________________________
Solution ________________________
Solve Algebraically: Choose substitution or elimination
3. y = 3x + 4
y = 4x
4. -3x + 6y = 12
x + 2y = 5
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5. y = 2x + 1
2y = 4x + 2
7.6 – Writing and Using a Linear System in Real Life
General Set-Up:
x + y = __________
___x + ___y = ________
$1590 was collected from 321 people at a museum. Adult tickets are $6 and child
tickets are $4. How many adults and how many children went into the museum?
1) Define your variables.
2) Write your system by writing two different equations.
3) Solve:
4) Answer the question asked.
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Example 1:
An office supply company sells two types of fax machines. They charge $150 for
one of the machines and $225 for the other. If the company sold 22 fax machines
for a total of $3900 last month, how many of each type were sold?
1) Define your variables.
2) Write your system by writing two different equations.
3) Solve:
4) Answer the question asked.
Example 2:
Your math teacher tells you that next week’s test is worth 100 points and contains 38
problems. Each problem is either worth 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?
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Example 3:
Write and solve a system of equations that represents each situation. Elain bought a total of 15
shirts and Paris of pants. She bought 7 more shirts than pants. How many of each did she buy?
Example 4
Eight times a number plus five times another number is –13. The sum of the two numbers is 1.
What are the numbers?
Example 5
Two times a number plus three times another number equals 4. Three times the first number plus
four times the other number is 7. Find the numbers.
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PRACTICE
1) Two times a number plus three times another number equals 13. The sum of the two numbers
is 7. What are the numbers?
2) Four times a number minus twice another number is –16. The sum of the two numbers is –1.
Find the numbers.
3) Two times a number plus three times another number equals 4. Three times the first number
plus four times the other number is 7. Find the numbers.
4) The cost of 8 muffins and 2 quarts of milk is $18. The cost of 3 muffins and 1 quart of milk is
$7.50. How much does 1 muffin and 1 quart of milk cost? Write an equation that represents
the situation. Then solve a system of equation using the elimination method.
5) FUNDRAISING Trisha and Byron are washing and vacuuming cars to raise money for a class
trip. Trisha raised $38 washing 5 cars and vacuuming 4 cars. Byron raised $28 by washing 4
cars and vacuuming 2 cars. Find the amount they charged to wash a car and vacuum a car.
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6. Gregory’s Motorsports has motorcycles (two wheels) and ATVs (four wheels) in stock. The
store has a total of 45 vehicles that, together, have 130 wheels.
a. Write a system of equations that represents the situation.
b. Solve the system of equations and interpret the solution.
7. Creative Crafts gives scrapbooking lessons for $15 per hour plus a $10 supply charge.
Scrapbooks Incorporated gives lessons for $20 per hour with no additional charges.
Write and solve a system of equations that represents the situation. Interpret the solution.
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