6-2 Substitution
Use substitution to solve each system of equations.
1. y = x + 5
3x + y = 25
SOLUTION: y=x+5
3x + y = 25
Substitute x + 5 for y in the second equation.
Substitute the solution for x into either equation to find y.
The solution is (5, 10).
2. x = y − 2
4x + y = 2
SOLUTION: x =y − 2
4x + y = 2
Substitute y − 2 for x in the second equation.
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Use the solution for y and either equation to find x.
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6-2 Substitution
The solution is (5, 10).
2. x = y − 2
4x + y = 2
SOLUTION: x =y − 2
4x + y = 2
Substitute y − 2 for x in the second equation.
Use the solution for y and either equation to find x.
The solution is (0, 2).
3. 3x + y = 6
4x + 2y = 8
SOLUTION: 3x + y = 6
4x + 2y = 8
First, solve the first equation for y to get y = −3x + 6. Then substitute −3x + 6 for y in the second equation.
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the solution
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and either
equation to find y.
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6-2 Substitution
The solution is (0, 2).
3. 3x + y = 6
4x + 2y = 8
SOLUTION: 3x + y = 6
4x + 2y = 8
First, solve the first equation for y to get y = −3x + 6. Then substitute −3x + 6 for y in the second equation.
Use the solution for x and either equation to find y.
The solution is (2, 0).
4. 2x + 3y = 4
4x + 6y = 9
SOLUTION: 2x + 3y = 4
4x + 6y = 9
First, solve the first equation for x.
Then substitute
for x in the second equation.
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6-2 Substitution
The solution is (2, 0).
4. 2x + 3y = 4
4x + 6y = 9
SOLUTION: 2x + 3y = 4
4x + 6y = 9
First, solve the first equation for x.
Then substitute
for x in the second equation.
Since the left side does not equal the right, there is no solution to this system of equations.
5. x − y = 1
3x = 3y + 3
SOLUTION: x −y = 1
3x = 3y + 3
First, solve the first equation for x to get x = y + 1. Then substitute y + 1 for x in the second equation.
This equation is an identity. So, there are infinitely many solutions.
6. 2x − y = 6
−3y = −6x + 18
SOLUTION: 2x − y = 6
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−3y = −6x + 18
First, solve the first equation for y to get 2x − 6 = y. Then substitute 2x − 6 for y in the second equation.
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6-2 Substitution
This equation is an identity. So, there are infinitely many solutions.
6. 2x − y = 6
−3y = −6x + 18
SOLUTION: 2x − y = 6
−3y = −6x + 18
First, solve the first equation for y to get 2x − 6 = y. Then substitute 2x − 6 for y in the second equation.
This equation is an identity. So, there are infinitely many solutions.
7. GEOMETRY The sum of the measures of angles X and Y is 180°. The measure of angle X is 24° greater than the measure of angle Y.
a. Define the variables, and write equations for this situation.
b. Find the measure of each angle.
SOLUTION: a. Let x = m X, and y = m Y; The sum means to add so the first equation is x + y = 180. Greater than means
addition as well, so the second equation is x = 24 + y.
b. x + y = 180
x = 24 + y
Substitute the second equation into the first equation.
Use the solution for y and either equation to find x.
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So, m
X = 102° and m
Y = 78°.
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6-2 Substitution
This equation is an identity. So, there are infinitely many solutions.
7. GEOMETRY The sum of the measures of angles X and Y is 180°. The measure of angle X is 24° greater than the measure of angle Y.
a. Define the variables, and write equations for this situation.
b. Find the measure of each angle.
SOLUTION: a. Let x = m X, and y = m Y; The sum means to add so the first equation is x + y = 180. Greater than means
addition as well, so the second equation is x = 24 + y.
b. x + y = 180
x = 24 + y
Substitute the second equation into the first equation.
Use the solution for y and either equation to find x.
So, m
X = 102° and m
Y = 78°.
Use substitution to solve each system of equations.
9. y = 4x + 5
2x + y = 17
SOLUTION: y = 4x + 5
2x + y = 17
Substitute 4x + 5 for y in the second equation.
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6-2 Substitution
So, m X = 102° and m
Y = 78°.
Use substitution to solve each system of equations.
9. y = 4x + 5
2x + y = 17
SOLUTION: y = 4x + 5
2x + y = 17
Substitute 4x + 5 for y in the second equation.
Use the solution for x and either equation to find y.
The solution is (2, 13).
11. y = 3x − 2
y = 2x − 5
SOLUTION: y = 3x − 2
y = 2x − 5
Substitute 2x − 5 for y in the first equation.
Use the solution for x and either equation to find y.
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6-2 Substitution
The solution is (2, 13).
11. y = 3x − 2
y = 2x − 5
SOLUTION: y = 3x − 2
y = 2x − 5
Substitute 2x − 5 for y in the first equation.
Use the solution for x and either equation to find y.
The solution is (−3, −11).
13. 3x + 4y = −3
x + 2y = −1
SOLUTION: 3x + 4y = −3
x + 2y = −1
First, solve the second equation for x to get x = −2y − 1. Then, substitute −2y − 1 for x in the first equation.
Use the solution for y and either equation to find x.
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6-2 Substitution
The solution is (−3, −11).
13. 3x + 4y = −3
x + 2y = −1
SOLUTION: 3x + 4y = −3
x + 2y = −1
First, solve the second equation for x to get x = −2y − 1. Then, substitute −2y − 1 for x in the first equation.
Use the solution for y and either equation to find x.
The solution is (−1, 0).
15. −1 = 2x − y
8x − 4y = −4
SOLUTION: −1 = 2x − y
8x − 4y = −4
First, solve the first equation for y to get 1 + 2x = y. Then substitute 1 + 2x for y in the second equation.
This equation is an identity. So, there are infinitely many solutions.
17. y = −4x + 11
3x + y = 9
SOLUTION: y = −4x + 11
3x + y = 9
Substitute −4x + 11 for y in the second equation.
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6-2 Substitution
This equation is an identity. So, there are infinitely many solutions.
17. y = −4x + 11
3x + y = 9
SOLUTION: y = −4x + 11
3x + y = 9
Substitute −4x + 11 for y in the second equation.
Use the solution for x and either equation to find y.
The solution is (2, 3).
19. 3x + y = −5
6x + 2y = 10
SOLUTION: 3x + y = −5
6x + 2y = 10
First, solve the first equation for y to get y = −3x −5. Then substitute −3x −5 for y in the second equation.
Since the left side does not equal the right, there is no solution to this system of equations.
21. 2x + y = 4
−2x + y = −4
SOLUTION: 2x + y = 4
−2x + y = −4
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First, solve the first equation for y to get y = −2x + 4. Then, substitute −2x + 4 for y in the second equation.
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6-2 Substitution
Since the left side does not equal the right, there is no solution to this system of equations.
21. 2x + y = 4
−2x + y = −4
SOLUTION: 2x + y = 4
−2x + y = −4
First, solve the first equation for y to get y = −2x + 4. Then, substitute −2x + 4 for y in the second equation.
Use the solution for x and either equation to find y.
The solution is (2, 0).
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