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Name
Class
6-1
Date
Reteaching
Solving Systems by Graphing
Graphing is useful for solving a system of equations. Graph both equations and look
for a point of intersection, which is the solution of that system. If there is no point of
intersection, there is no solution.
Problem
What is the solution to the system? Solve by graphing. Check.
x+y=4
2x – y = 2
Solution
y = –x + 4
y = 2x – 2
Put both equations into y-intercept form, y = mx + b.
y = –x + 4
The first equation has a y-intercept of (0, 4).
0 = –x + 4
Find a second point by substituting in 0 for y and solve for x.
x=4
You have a second point (4, 0), which is the x-intercept.
y = 2x – 2
The second equation has a y-intercept of (0, –2).
0 = 2(x) – 2
Find a second point by substituting in 0 for y and solve for x.
2 = 2x, x = 1
You have a second point for the second line, (1, 0).
2 = 2x, x = 1
You have a second point for the second line, (1, 0).
Plot both sets of points and draw both lines. The lines appear to
intersect (2, 2), so (2, 2) is the solution.
Check
If you substitute in the point (2, 2), for x and y in your original equations, you can
double-check your answer.
x+y=4
2+2
4,
4=4
2x – y = 2
2(2) – 2
2,
2=2
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Name
6-1
Class
Date
Reteaching (continued)
Solving Systems by Graphing
If the equations represent the same line, there is an infinite number of
solutions, the coordinates of any of the points on the line.
Problem
What is the solution to the system? Solve by graphing. Check.
2x – 3y = 6
4x – 6y = 18
Solution
What do you notice about these equations? Using the y-intercepts and
solving for the x-intercepts, graph both lines using both sets of points.
2
x2
3
2
y  x 3
3
y
Graph equation 1 by finding two points: (0, –2) and (3, 0). Graph
equation 2 by finding two points (0, –3) and (4.5, 0).
Is there a solution? Do the lines ever intersect? Lines with the
same slope are parallel. Therefore, there is no solution to this
system of equations.
Exercises
Solve each system of equations by graphing. Check.
1. 2x = 2 – 9y
21y = 4 – 6x
2. 2x = 3 – y
3. y = 1.5x + 4
y = 4x – 12
4. 6y = 2x – 14
5. 3y = –6x – 3
x – 7 = 3y
y = 2x – 1
7. 2x + 3y = 11
8. 3y = 3x – 6
x – y = –7
y=x–2
0.5x + y = –2
6. 2x = 3y – 12
1x
3
= 4y + 5
9. y = 12 x + 9
2y – x = 1
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Name
Class
Date
Reteaching
6-2
Solving Systems Using Substitution
You can solve a system of equations by substituting an equivalent expression for
one variable.
Problem
Solve and check the following system:
x + 2y = 4
2x – y = 3
x + 2y = 4
Solution
x = 4 – 2y
2(4 – 2y) –y = 3
8 – 4y – y = 3
Get x to one side by subtracting 2y.
Substitute 4 – 2y for x in the second equation.
Distribute.
Simplify.
8 – 5y = 3
8 – 8 – 5y =3 –8
–5y = –5
Subtract 8 from both sides.
Divide both sides by 25.
y=1
You have the solution for y. Solve for x.
x +2(1) = 4
Substitute in 1 for y in the first equation.
x+2–2=4–2
x=2
Check
The first equation is easiest to solve in terms of one
variable.
Subtract 2 from both sides.
The solution is (2, 1).
Substitute your solution into either of the given linear equations.
x + 2y = 4
2 + 2(1)
4
4=4
Substitute (2, 1) into the first equation.
You check the second equation.
Exercises
Solve each system using substitution. Check your answer.
1. x + y = 3
2. x – 3y = –14
2x – y = 0
x–y = –2
3. 2x – 2y =10
4. 4x + y = 8
x + 2y = 5
x–y=5
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Reteaching (continued)
6-2
Solving Systems Using Substitution
Problem
Solve and check the following system:
x
 3y  10
2
3x + 4y = –6
First, isolate x in the first equation.
x
– 3y = 10
2
Solve
x
2
Add 3y to both sides and simplify.
= 10 + 3y
Multiply by 2 on both sides.
x = 20 + 6y
Substitute 20 + 6y for x in second equation.
3x + 4y = – 6
Simplify.
3(20 + 6y) + 4y = –6
Subtract 60 from both sides.
60 + 22y = – 6
Divide by 22 to solve for y.
22y = –66, y = – 3
x
2
Substitute 23 in the first equation.
– 3(–3) = 10
x
2
Simplify.
+ 9 = 10
Solve for x.
x=2
The solution is (2, –3).
Check
3(2) + 4(–3) 26
–6 = –6
Now you check the first equation.
Exercises
Solve each system using substitution. Check your answer.
5. –2x + y = 8
3x + y = –2
6. 3x – 4y = 8
2x + y = 9
7. 3x + 2y = 25
2x + 3y = –6
8. 6x – 5y = 3
x – 9y = 25
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Name
Class
Date
Reteaching
6-3
Solving Systems Using Elimination
Elimination is one way to solve a system of equations. Think about what the
word “eliminate” means. You can eliminate either variable, whichever is easiest.
Problem
4x – 3y = –4
Solve and check the following system of linear equations.
2x + 3y = 34
Solution The equations are already arranged so that like terms are in columns.
Notice how the coefficients of the y-variables have the opposite sign and the same
value.
4x – 3y = –4
2x + 3y = 34
6x = 30
Add the equations to eliminate y.
Divide both sides by 6 to solve for x.
x=5
4(5) – 3y = –4
20 – 3y = –4
–3y = –24
Substitute 5 for x in one of the original equations
and solve for y.
y=8
The solution is (5, 8).
Check
4x – 3y = –4
4(5) – 3(8) –4
20 – 24 –4
–4 = –4
Substitute your solution into both of
the original equations to check.
You can check the other equaton.
Exercises
Solve and check each system.
1. 3x + y = 3
2. 6x – 3y = –14
–3x + y = 3
6x – y = –2
3. 3x – 2y = 10
4. 4x + y = 8
x – 2y = 6
x+y=5
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Reteaching (continued)
6-3
Solving Systems Using Elimination
If none of the variables has the same coefficient, you have to multiply before
you eliminate.
Problem
–2x + 3y = –1
Solve the following system of linear equations.
5x + 4y = 6
Solution
5(–2x – 3y) = (–1)5
2(5x + 4y) = (6)2
Multiply the first equation by 5 (all terms, both sides)
and the second equation by 2. You can eliminate the x
variable when you add the equations together.
–10x – 15y = –5
Distribute, simplify and add.
10x + 8y = 12
–7y = 7
y = –1
5x + 4(–1) = 6
Divide both sides by 7.
Substitute –1 in for y in the second equation to find the
value of x.
Simplify.
5x – 4 = 6
Add 4 to both sides.
5x = 10
x=2
Divide by 5 to solve for x.
The solution is (2, –1).
Check
–2x + 3y = –1
Substitute your solution into both original equations.
?
–2(2) – 3(–1)  –1
–1 = –1
You can check the other equation.
Exercises
Solve and check each system.
5.
6. –2x – 6y = 0
x – 3y = –3
–2x + 7y = 10
3x + 11y = 4
7. 3x + 10y = 5
8. 4x + y = 8
7x + 20y = 11
x+y=5
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Name
6-6
Class
Date
Reteaching
Systems of Linear Inequalities
A system of linear inequalities is a set of linear inequalities in the same plane. The
solution of the system is the region where the solution regions of the inequalities of
the system overlap.
Problem
What is the graph of the system of linear inequalities:
x – y > –1
y ≤ 2x + 3
?
Put the first inequality into slope-intercept form, y < x + 1. Use a dashed line since <
does not include the points on the boundary line in the solution. Using the point (0, 0),
decide where to shade the first inequality. The point (0, 0) makes the inequality true,
so shade the region including (0, 0).
Then graph the boundary line of the second inequality, y ≤ 2x + 3. It is a solid line
because of the ≤ sign. Use the point (0, 0) to decide where to shade the second
inequality. The point (0, 0) makes the second inequality true, so shade the region
including (0, 0).
The overlapping region of the 2 inequalities is the
solution to the system. It includes the points (0, 0), (1,
1), (3, 1). You can test any point in the region in both
equations to see if it makes both equations true. In
word problems, the solutions often cannot be negative
(cars, tickets sold, etc.). Two requirements are that x ≥
0 and y ≥ 0. Keep this in mind when graphing word
problems.
Problem
A cash register has fewer than 200 dimes and quarters worth more than $39.95.
How many of each coin are in the register?
The system of inequalities that you get from the table is:
q + d < 200
25q + 10d > 3995
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Reteaching (continued)
Systems of Linear Inequalities
Using elimination, solve for q by multiplying all terms in the first equation by –10 and
eliminating d: (q + d < 200)(–10).
–10q – 10d > –2000
25q + 10d > 3995
Now add the 2 systems together to solve for q.
15q > 1995
q > 133
q + d < 200
Write first inequality.
133 + d < 200, d < 67
Substitute in 133 for q, subtract 133 from both sides
and solve for d.
The register contains at least 133 quarters and no more than 67 dimes.
Exercises
Graph the following systems of inequalities.
1. x – 2y < 3
y
> 3x + 6
2
x
4. 3y ≥ 4
–y ≤ x + 2
2. y ≥ –x + 5
–x ≤ –2y – 3
3. x + 3y ≥ –4
3x – 2y < 5
5. 2x – y < 1
6. 5x – 4y ≥ 3
x + 2y < –4
2x + 3y ≤ –2
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Extra Practice
Chapter 6
Lesson 6-1
Solve each system by graphing.
1. x– y = 7
3x + 2y = 6
2. y = 2x + 3
3. y = –2x + 6
y  3 x4
2
3x + 4y = 24
Write and solve a system of equations by graphing.
4. One calling card has a $.50 connection fee and charges $.02 per minute.
Another card has a $.25 connection fee and charges $.03 per minute.
After how many minutes would a call cost the same amount using either
card?
5. Suppose that you have $75 in your savings account and you save an
additional $5 per week. Your friend has $30 in his savings account and saves
an additional $10 per week. In how many weeks will you both have the same
amount of money in your accounts?
Lesson 6-2
Solve each system by using substitution.
6. x – y = 13
y – x = –13
7. 3x – y = 4
x + 5y = –4
8. x + y = 4
y = 7x + 4
Write and solve a system of equations by substitution.
9. A farmer grows corn and soybeans on her 300-acre farm. She wants to
plant 110 more acres of soybeans than corn. How many acres of each
crop does she need to plant?
10. The perimeter of a rectangle is 34 cm. The length is 1 cm longer than the
width. What are the dimensions of the rectangle?
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Extra Practice (continued)
Chapter 6
Lesson 6-3
Solve each system by elimination.
11. x + y = 19
12. –3x + 4y = 29
x – y = –7
3x + 2y = –17
14. 6x + y = 13
15. 4x – 9y = 61
y – x = –8
13. 3x + y = 3
–3x + 2y = –30
16. 4x – y = 105
10x + 3y = 25
x + 7y = –10
Write and solve a system of equations using elimination.
17. Two groups of people order food at a restaurant. One group orders
4 hamburgers and 7 chicken sandwiches for $34.50. The other group orders
8 hamburgers and 3 chicken sandwiches for $30.50. Find the cost of each
item.
18. The sum of two numbers is 25. Their difference is 9. What are the two numbers?
Lesson 6-6
Solve each system by graphing.
38. y ≤ 5x + 1
y>x–3
41. y < –2x + 1
y > –2x – 3
39. y > 4x + 3
40. y > –x + 2
y ≥ –2x – 1
42. y ≤ 5
y>x–4
43. y ≤ 5x – 2
y ≥ – x +1
y>3
44. Hideo plans to spend no more than $60 at an entertainment store on DVDs and CDs.
DVDs cost $17 each and CDs cost $14 each. He wants to buy at least two items.
Write and graph a system of linear inequalities that describes the situation. What are
three possible combinations of CDs and DVDs that he can buy? Write and graph a
system of inequalities that describes the situation.
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