Solving Systems by Elimination

4.3. SOLVING SYSTEMS BY ELIMINATION
243
47. The second equation, y = −3 − 2x, is already solved for y. Substitute
−3 − 2x for y in the first equation and solve for x.
−9x + 4y = 73
−9x + 4(−3 − 2x) = 73
First Equation.
Substitute −3 − 2x for y.
−9x − 12 − 8x = 73
−12 − 17x = 73
Distribute the 4.
Combine like terms.
−17x = 85
x = −5
Add 12 to both sides.
Divide both sides by −17.
Finally, to find the y-value, substitute −5 for x in the equation
y = −3 − 2x.
y = −3 − 2x
y = −3 − 2(−5)
y = −3 + 10
Substitute −5 for x.
Multiply.
y=7
Simplify.
Hence, (x, y) = (−5, 7) is the solution of the system.
4.3
Solving Systems by Elimination
1. Start with the given system.
x +
9x −
4y
7y
=
=
0
−43
We’ll concentrate on eliminating x. Multiply the first equation by −9, then
add the results.
−9x − 36y =
0
9x − 7y = −43
− 43y
=
−43
Divide both sides by −43.
−43
−43
y=1
y=
Divide both sides by −43.
Simplify.
Take the answer y = 1 and substitute 1 for y in the first equation.
x + 4y = 0
x + 4(1) = 0
x+4=0
x = −4
First equation.
Substitute 1 for y.
Multiply.
Subtract 4 from both sides.
Hence, the solution is (x, y) = (−4, 1).
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CHAPTER 4. SYSTEMS
244
Check: We must show that the solution (x, y) = (−4, 1) satisfies both equations.
x + 4y = 0
9x − 7y = −43
(−4) + 4(1) = 0
−4 + 4 = 0
9(−4) − 7(1) = −43
−36 − 7 = −43
0=0
−43 = −43
Because each of the last two statements are true, this guarantees that (x, y) =
(−4, 1) is a solution of the system.
3. Start with the given system.
6x + y
4x + 2y
=
=
8
0
We’ll concentrate on eliminating y. Multiply the first equation by −2, then
add the results.
−12x − 2y = −16
4x + 2y =
0
−8x
=
−16
Divide both sides by −8.
−16
−8
x=2
x=
Divide both sides by −8.
Simplify.
Take the answer x = 2 and substitute 2 for x in the first equation.
6x + y = 8
First equation.
6(2) + y = 8
12 + y = 8
Substitute 2 for x.
Multiply.
y = −4
Subtract 12 from both sides.
Hence, the solution is (x, y) = (2, −4).
Check: We must show that the solution (x, y) = (2, −4) satisfies both equations.
6x + y = 8
4x + 2y = 0
6(2) + (−4) = 8
12 − 4 = 8
4(2) + 2(−4) = 0
8−8=0
8=8
0=0
Second Edition: 2012-2013
4.3. SOLVING SYSTEMS BY ELIMINATION
245
Because each of the last two statements are true, this guarantees that (x, y) =
(2, −4) is a solution of the system.
5. Start with the given system.
−8x +
4x +
y
3y
=
=
−56
56
We’ll concentrate on eliminating y. Multiply the first equation by −3, then
add the results.
24x − 3y = 168
4x + 3y = 56
28x
=
224
Divide both sides by 28.
224
28
x=8
Divide both sides by 28.
x=
Simplify.
Take the answer x = 8 and substitute 8 for x in the first equation.
−8x + y = −56
−8(8) + y = −56
First equation.
Substitute 8 for x.
−64 + y = −56
y=8
Multiply.
Add 64 to both sides.
Hence, the solution is (x, y) = (8, 8).
Check: We must show that the solution (x, y) = (8, 8) satisfies both equations.
−8x + y = −56
−8(8) + (8) = −56
4x + 3y = 56
4(8) + 3(8) = 56
−64 + 8 = −56
32 + 24 = 56
−56 = −56
56 = 56
Because each of the last two statements are true, this guarantees that (x, y) =
(8, 8) is a solution of the system.
7. Start with the given system.
x +
−5x −
8y
9y
=
=
41
−50
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CHAPTER 4. SYSTEMS
246
We’ll concentrate on eliminating x. Multiply the first equation by 5, then add
the results.
5x + 40y = 205
−5x − 9y = −50
31y
=
155
Divide both sides by 31.
155
31
y=5
Divide both sides by 31.
y=
Simplify.
Take the answer y = 5 and substitute 5 for y in the first equation.
x + 8y = 41
First equation.
x + 8(5) = 41
x + 40 = 41
Substitute 5 for y.
Multiply.
x=1
Subtract 40 from both sides.
Hence, the solution is (x, y) = (1, 5).
Check: We must show that the solution (x, y) = (1, 5) satisfies both equations.
x + 8y = 41
−5x − 9y = −50
(1) + 8(5) = 41
1 + 40 = 41
−5(1) − 9(5) = −50
−5 − 45 = −50
41 = 41
−50 = −50
Because each of the last two statements are true, this guarantees that (x, y) =
(1, 5) is a solution of the system.
9. Start with the given system.
−12x + 9y
−6x − 4y
=
=
0
−34
We’ll concentrate on eliminating x. Multiply the second equation by −2, then
add the results.
−12x + 9y =
0
12x + 8y = 68
17y
Second Edition: 2012-2013
=
68
4.3. SOLVING SYSTEMS BY ELIMINATION
247
Divide both sides by 17.
17y = 68
68
y=
17
y=4
Divide both sides by 17.
Simplify.
Take the answer y = 4 and substitute 4 for y in the first equation.
−12x + 9y = 0
−12x + 9(4) = 0
First equation.
Substitute 4 for y.
−12x + 36 = 0
−12x = −36
Multiply.
Subtract 36 from both sides.
Divide both sides by −12
x=3
Hence, the solution is (x, y) = (3, 4).
11. Start with the given system.
27x −
−3x −
6y
5y
=
=
−96
22
We’ll concentrate on eliminating x. Multiply the second equation by 9, then
add the results.
27x −
6y = −96
−27x − 45y = 198
−
51y
=
102
Divide both sides by −51.
−51y = 102
102
y=
−51
y = −2
Divide both sides by −51.
Simplify.
Take the answer y = −2 and substitute −2 for y in the first equation.
27x − 6y = −96
27x − 6(−2) = −96
27x + 12 = −96
27x = −108
x = −4
First equation.
Substitute −2 for y.
Multiply.
Subtract 12 from both sides.
Divide both sides by 27
Hence, the solution is (x, y) = (−4, −2).
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CHAPTER 4. SYSTEMS
248
13. Start with the given system.
2x −
−3x +
6y
18y
=
=
28
−60
We’ll concentrate on eliminating y. Multiply the first equation by 3, then add
the results.
6x − 18y =
84
−3x + 18y = −60
3x
=
24
Divide both sides by 3.
3x = 24
24
x=
3
x=8
Divide both sides by 3.
Simplify.
Take the answer x = 8 and substitute 8 for x in the first equation.
2x − 6y = 28
2(8) − 6y = 28
First equation.
Substitute 8 for x.
16 − 6y = 28
−6y = 12
Multiply.
Subtract 16 from both sides.
y = −2
Divide both sides by −6
Hence, the solution is (x, y) = (8, −2).
15. Start with the given system.
−32x + 7y
8x − 4y
= −238
=
64
We’ll concentrate on eliminating x. Multiply the second equation by 4, then
add the results.
−32x + 7y = −238
32x − 16y =
256
−
9y
=
18
Divide both sides by −9.
−9y = 18
18
y=
−9
y = −2
Second Edition: 2012-2013
Divide both sides by −9.
Simplify.
4.3. SOLVING SYSTEMS BY ELIMINATION
249
Take the answer y = −2 and substitute −2 for y in the first equation.
−32x + 7y = −238
−32x + 7(−2) = −238
First equation.
Substitute −2 for y.
−32x − 14 = −238
−32x = −224
Multiply.
Add 14 to both sides.
Divide both sides by −32
x=7
Hence, the solution is (x, y) = (7, −2).
17. Start with the given system.
3x −
−2x −
7y
2y
=
=
−75
−10
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by 2, the second equation by 3, then add the results.
6x −
−6x −
14y
6y
=
=
−150
−30
−
20y
=
−180
Divide both sides by −20.
−20y = −180
−180
y=
−20
y=9
Divide both sides by −20.
Simplify.
Take the answer y = 9 and substitute 9 for y in the first equation (you could
also make the substitution in the second equation).
3x − 7y = −75
3x − 7(9) = −75
3x − 63 = −75
3x = −12
−12
x=
3
x = −4
First equation.
Substitute 9 for y.
Multiply.
Add 63 to both sides.
Divide both sides by 3.
Simplify.
Hence, the solution is (x, y) = (−4, 9).
Second Edition: 2012-2013
CHAPTER 4. SYSTEMS
250
19. Start with the given system.
9x − 9y
2x − 6y
=
=
−63
−34
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by −2, the second equation by 9, then add the results.
−18x +
18x −
18y
54y
=
=
126
−306
−
36y
=
−180
Divide both sides by −36.
−36y = −180
−180
y=
−36
y=5
Divide both sides by −36.
Simplify.
Take the answer y = 5 and substitute 5 for y in the first equation (you could
also make the substitution in the second equation).
9x − 9y = −63
First equation.
9x − 9(5) = −63
9x − 45 = −63
Substitute 5 for y.
Multiply.
9x = −18
−18
x=
9
x = −2
Add 45 to both sides.
Divide both sides by 9.
Simplify.
Hence, the solution is (x, y) = (−2, 5).
21. Start with the given system.
−9x − 2y
5x − 3y
=
28
= −32
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by −5, the second equation by −9, then add the results.
45x +
−45x +
Second Edition: 2012-2013
10y
27y
=
=
−140
288
37y
=
148
4.3. SOLVING SYSTEMS BY ELIMINATION
251
Divide both sides by 37.
37y = 148
148
y=
37
y=4
Divide both sides by 37.
Simplify.
Take the answer y = 4 and substitute 4 for y in the first equation (you could
also make the substitution in the second equation).
−9x − 2y = 28
−9x − 2(4) = 28
First equation.
Substitute 4 for y.
−9x − 8 = 28
−9x = 36
36
x=
−9
x = −4
Multiply.
Add 8 to both sides.
Divide both sides by −9.
Simplify.
Hence, the solution is (x, y) = (−4, 4).
23. Start with the given system.
−3x −
7x +
5y
7y
=
=
−34
56
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by −7, the second equation by −3, then add the results.
21x + 35y
−21x − 21y
=
=
238
−168
14y
=
70
Divide both sides by 14.
14y = 70
70
y=
14
y=5
Divide both sides by 14.
Simplify.
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CHAPTER 4. SYSTEMS
252
Take the answer y = 5 and substitute 5 for y in the first equation (you could
also make the substitution in the second equation).
−3x − 5y = −34
−3x − 5(5) = −34
First equation.
Substitute 5 for y.
−3x − 25 = −34
−3x = −9
−9
x=
−3
x=3
Multiply.
Add 25 to both sides.
Divide both sides by −3.
Simplify.
Hence, the solution is (x, y) = (3, 5).
25. Start with the given system.
2x −
7x +
7y
6y
−2
3
=
=
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by −7, the second equation by 2, then add the results.
−14x +
14x +
49y
12y
=
=
14
6
61y
=
20
Divide both sides by 61 to get y = 20/61. Next, we could substitute 20/61 for
y in either equation and solve to find x. However, in this case it is probably
easier to perform elimination again. Start with the given system again.
2x −
7x +
7y
6y
−2
3
=
=
This time we concentrate on eliminating the variable y. Multiply the first
equation by −6, the second equation by −7, then add the results.
−12x + 42y
−49x − 42y
=
=
12
−21
−61x
=
−9
Divide both sides by −61 to get x = 9/61. Hence, the solution is (x, y) =
(9/61, 20/61).
Check: First, store 9/61 in X with the following keystrokes. The result is
shown in the first image below.
9
÷
6
Second Edition: 2012-2013
1
STO X, T, θ, n
ENTER
4.3. SOLVING SYSTEMS BY ELIMINATION
253
Store 20/61 in Y with the following keystrokes. The result is shown in the first
image below.
2
0
÷
6
1
STO ALPHA
1
ENTER
Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is
shown in the second image below.
2
×
X, T, θ, n
−
×
7
ALPHA
1
ENTER
Enter the right-hand side of the second equation with the following keystrokes.
The result is shown in the second image below.
7
×
X, T, θ, n
+
×
6
ALPHA
1
ENTER
The result in the second image shows that 2x − 7y = −2 and 7x + 6y = 3 for
x = 9/61 and y = 20/61. The solution checks.
27. Start with the given system.
2x +
−5x +
3y
5y
=
=
−2
2
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by 5, the second equation by 2, then add the results.
10x +
−10x +
15y
10y
= −10
=
4
25y
=
−6
Divide both sides by 25 to get y = −6/25. Next, we could substitute −6/25
for y in either equation and solve to find x. However, in this case it is probably
easier to perform elimination again. Start with the given system again.
2x +
−5x +
3y
5y
=
=
−2
2
Second Edition: 2012-2013
CHAPTER 4. SYSTEMS
254
This time we concentrate on eliminating the variable y. Multiply the first
equation by −5, the second equation by 3, then add the results.
−10x −
−15x +
15y
15y
−25x
=
=
10
6
=
16
Divide both sides by −25 to get x = −16/25. Hence, the solution is (x, y) =
(−16/25, −6/25).
Check: First, store −16/25 in X with the following keystrokes. The result is
shown in the first image below.
(-)
1
÷
6
2
5
STO X, T, θ, n
ENTER
Store −6/25 in Y with the following keystrokes. The result is shown in the
first image below.
(-)
6
÷
2
5
STO ALPHA
1
ENTER
Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is
shown in the second image below.
×
2
X, T, θ, n
+
×
3
ALPHA
1
ENTER
Enter the right-hand side of the second equation with the following keystrokes.
The result is shown in the second image below.
(-)
5
×
X, T, θ, n
+
5
×
ALPHA
1
ENTER
The result in the second image shows that 2x + 3y = −2 and −5x + 5y = 2 for
x = −16/25 and y = −6/25. The solution checks.
Second Edition: 2012-2013
4.3. SOLVING SYSTEMS BY ELIMINATION
255
29. Start with the given system.
9x +
−7x −
4y
9y
−4
3
=
=
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by 7, the second equation by 9, then add the results.
63x +
−63x −
28y
81y
= −28
=
27
−
53y
=
−1
Divide both sides by −53 to get y = 1/53. Next, we could substitute 1/53 for
y in either equation and solve to find x. However, in this case it is probably
easier to perform elimination again. Start with the given system again.
9x +
−7x −
4y
9y
−4
3
=
=
This time we concentrate on eliminating the variable y. Multiply the first
equation by 9, the second equation by 4, then add the results.
81x +
−28x −
= −36
=
12
36y
36y
= −24
53x
Divide both sides by 53 to get x = −24/53. Hence, the solution is (x, y) =
(−24/53, 1/53).
Check: First, store −24/53 in X with the following keystrokes. The result is
shown in the first image below.
(-)
2
÷
4
5
3
STO X, T, θ, n
ENTER
Store 1/53 in Y with the following keystrokes. The result is shown in the first
image below.
1
÷
5
3
STO ALPHA
1
ENTER
Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is
shown in the second image below.
9
×
X, T, θ, n
+
4
×
ALPHA
1
ENTER
Enter the right-hand side of the second equation with the following keystrokes.
The result is shown in the second image below.
Second Edition: 2012-2013
CHAPTER 4. SYSTEMS
256
(-)
×
7
−
X, T, θ, n
9
×
ALPHA
1
ENTER
The result in the second image shows that 9x + 4y = −4 and −7x − 9y = 3 for
x = −24/53 and y = 1/53. The solution checks.
31. Start with the given system.
2x + 2y
3x − 5y
=
=
4
3
We’ll first concentrate on eliminating the variable x. Multiply the first equation
by −3, the second equation by 2, then add the results.
−6x −
6x −
6y
10y
=
=
−12
6
−
16y
=
−6
Divide both sides by −16 to get y = 3/8. Next, we could substitute 3/8 for
y in either equation and solve to find x. However, in this case it is probably
easier to perform elimination again. Start with the given system again.
2x + 2y
3x − 5y
=
=
4
3
This time we concentrate on eliminating the variable y. Multiply the first
equation by 5, the second equation by 2, then add the results.
10x + 10y
6x − 10y
=
=
20
6
16x
=
26
Divide both sides by 16 to get x = 13/8. Hence, the solution is (x, y) =
(13/8, 3/8).
Check: First, store 13/8 in X with the following keystrokes. The result is
shown in the first image below.
1
3
÷
Second Edition: 2012-2013
8
STO X, T, θ, n
ENTER
4.3. SOLVING SYSTEMS BY ELIMINATION
257
Store 3/8 in Y with the following keystrokes. The result is shown in the first
image below.
3
÷
8
STO 1
ALPHA
ENTER
Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is
shown in the second image below.
2
×
X, T, θ, n
+
×
2
ALPHA
1
ENTER
Enter the right-hand side of the second equation with the following keystrokes.
The result is shown in the second image below.
3
×
X, T, θ, n
−
×
5
ALPHA
1
ENTER
The result in the second image shows that 2x + 2y = 4 and 3x − 5y = 3 for
x = 13/8 and y = 3/8. The solution checks.
33. Start with the given system.
x + 7y
−8x − 56y
=
=
−32
256
We’ll concentrate on eliminating x. Multiply the first equation by 8, then add
the results.
8x + 56y = −256
−8x − 56y =
256
0 =
0
Note that this last statement, 0 = 0, is true. Hence, the system has an infinite
number of solutions.
Second Edition: 2012-2013
CHAPTER 4. SYSTEMS
258
35. Start with the given system.
16x −
−8x +
16y
8y
= −256
=
128
We’ll concentrate on eliminating x. Multiply the second equation by 2, then
add the results.
16x − 16y = −256
−16x + 16y =
256
0 =
0
Note that this last statement, 0 = 0, is true. Hence, the system has an infinite
number of solutions.
37. Start with the given system.
x − 4y
2x − 8y
=
=
−37
54
We’ll concentrate on eliminating x. Multiply the first equation by −2, then
add the results.
−2x + 8y = 74
2x − 8y = 54
0 =
128
Note that this last statement, 0 = 128, is false. Hence, the system has no
solution.
39. Start with the given system.
x + 9y
−4x − 5y
=
73
= −44
We’ll concentrate on eliminating x. Multiply the first equation by 4, then add
the results.
4x + 36y = 292
−4x − 5y = −44
31y
=
248
Divide both sides by 31.
31y = 248
248
y=
31
y=8
Second Edition: 2012-2013
Divide both sides by 31.
Simplify.