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Skill Builder 5.1
Solving Systems of Linear Equations by Graphing
Solve the following systems by graphing. State the intersection point as an ordered pair (x, y) or
state if no solution, or state if infinite number of solutions.
1. x + y = –4
x–y=2
2. 8x + 2y = 6
4x + y = 3
Solution: _______________
Solution: ________________
3. x – y = 3
x=4
4. 2x + y = 1
x–y=5
Solution: _______________
Solution: ________________
5. 3x – 2y = –4
–3x + 2y = –2
6. x + y = 5
x–y=1
Solution: _______________
Solution: __________________
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Skill Builder 5.2
Solving Systems of Equations by Substitution
Solving systems using the Substitution Method.
• Isolate x or y first. Choose the variable with a coefficient of 1 (if there is
one) to isolate.
• Substitute the value of the isolated letter into the equation that was not
changed.
• Once you have a value for 1 variable – back substitute to find the value
of the other variable.
• Remember… most systems have ordered pair solutions.
• If one of the variables “disappears” while solving, the system has either
no solution or infinite solutions.
Solve each system by the substitution method. Be sure to check your answers.
Show your check.
1. x + y = 2
x–y=6
2. 2x – 3y = 12
y = 2x – 8
3. 3x – 4y = –1
4x – y = 3
4. 10x – 5y = 20
x + 6y = –11
5. 4x + 2y = –3
2x + y = 1
6. x + 3y = 5
x–y=1
Is the ordered pair a solution to the system? Circle Yes or No.
1. 2x – 3y = 12
2x – y = 8
(–3, 2)
Yes or No
2. –x – y = –3
2x + y = 1
(–2, 5)
Yes or No
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3. 3x – y = 7
x = 7 – 2y
(3, 2)
Yes or No
Skill Builder 5.3
Solving Systems of Linear Equations by the Addition Method
For the following six linear systems, solve two by graphing using graphs below, two by addition,
and two by substitution. Evaluate the most efficient method for solving each system before
beginning.
1. y = –x
y = –x + 4
2.
3. x + y = 10
y=x+8
4. 3x – 2y = 10
5x + 3y = 4
5. y = 3
x=5
6. 2x + y = 7
x–y=8
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y=5–x
3x – 4y = –20
Skill Builder 5.4
Problem Solving
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Read, read and reread the problem.
Identify unknowns with variables.
Write a system of equations. 2 variables = 2 equations.
Solve the system.
Answer with words.
Use a system of equations to solve each of the following problems.
1. The sum of two numbers is 18. Find the two numbers if twice the smaller number is 6 more than
the larger.
2. The difference of two positive numbers is 28. One number is three times the other. Find the
numbers.
3. Grandma Betty is going to buy her grandchildren puppies for Christmas. She can buy 2 cocker
spaniels and 3 golden retrievers for $1200 or she can buy 3 cocker spaniels and 2 golden
retrievers for $1050. Find the cost of a cocker spaniel and a golden retriever.
4. A cable TV company is promoting two special deals. One deal includes basic cable service and
one movie channel for $35 per month. The other deal includes basic service with two movie
channels for $45 per month. Find the cost of basic cable and the charge for each movie channel.
5. Beth is saving dimes and quarters. She already has 3 times as many quarters as dimes. If the sum
of the number of dimes and twice the number of quarters is 21, how many quarters does she
have? How many dimes does she have? How much money does she have?
6. Jake rode his bike 4 miles farther than Mike. Together they rode a distance of 20 miles. How far
did each boy ride?
7. Carol bought 3 small steaks and 1 bag of potatoes at the grocery store for $11. Three days later
she returned to the store and purchased 2 steaks and 3 bags of potatoes for $12. How much does 1
steak cost? How much does a bag of potatoes cost?
8. A vacant lot has a perimeter of 680 feet. The lot is 60 feet longer than it is wide. Find the length
and width of the lot.
9. Jenna spent $175 for Christmas gifts for 23 nurses at the hospital. She bought candles for some
that cost $5 each and she bought scarves for the others that cost $10 each. How many candles did
she buy? How many scarves did she buy?
10. Macy has friends coming for the weekend. She can buy 2 new bath towels and 1 hand towel for
$29 or she can buy 3 new bath towels and 2 hand towels for $46. Find the cost of one bath towel.
11. A plane can travel 4200 miles in 6 hours with a tail wind. Flying against the wind, it takes the
plane 7 hours to make the return flight. Find the plane’s speed with no wind. Find the wind speed.
12. The local theater is having a Saturday afternoon performance. Adult tickets cost $5.50 and
children’s tickets cost $4.00. If 70 tickets are sold for a total of $310, how many adult tickets
were sold? How many children’s tickets were sold?
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Skill Builder 5.5
Systems of Linear Inequalities
Graph the solution of each system of linear inequality.
1. x > 1
y<2
2. y ≥ x
y<x+1
3. y > 4x – 1
y ≤ –2x +3
4. y < 2x
y≥x–5
5. x – 2y ≥ 6
x + 2y ≤ 4
6. 2x + y > 8
2x – y > 1
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