Solving Systems by Graphing

Chapter
4
Systems
4.1
Solving Systems by Graphing
1. First, determine the x- and y-intercepts of 3x − 4y = 24.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
3x − 4y = 24
3x − 4y = 24
3x − 4(0) = 24
3(0) − 4y = 24
3x = 24
x=8
−4y = 24
y = −6
Plot and label the intercepts, then draw the line 3x − 4y = 24 through them
and label it with its equation. Next, the line y = − 12 x − 1 has slope −1/2 and
y-intercept (0, −1). Plot and label (0, −1), then move 2 units to the right and
1 unit down. Label the resulting line with its equation.
y
y
3x − 4y = 24
2
(8, 0)
−2
10
x
2
Δx = 2
−2
(0, −1)
10
x
Δy = −1
(0, −6)
y = − 21 x − 1
−10
−10
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206
Next, place both lines on the same coordinate system, label each line with its
equation, then label the point of intersection with its coordinates.
y
3x − 4y = 24
2
−2
10
x
(4, −3)
y = − 12 x − 1
−10
Substitute the point (x, y) = (4, −3) in both equations to see if it checks.
1
y =− x−1
2
1
−3 = − (4) − 1
2
−3 = −3
3x − 4y = 24
3(4) − 4(−3) = 24
24 = 24
Hence, the solution (x, y) = (4, −3) checks.
3. First, determine the x- and y-intercepts of 2x + y = 6.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
2x + y = 6
2x + (0) = 6
2x + y = 6
2(0) + y = 6
2x = 6
x=3
y=6
Plot and label the intercepts, then draw the line 2x + y = 6 through them
and label it with its equation. Next, the line y = x + 3 has slope 1 and yintercept (0, 3). Plot and label (0, 3), then move 1 unit to the right and 1 unit
up. Label the resulting line with its equation.
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207
y
y
10
y = x+3
10
(0, 6)
(0, 3)
Δy = 1
Δx = 1
(3, 0)
−2
10
−2
x
−2
10
−2
2x + y = 6
Next, place both lines on the same coordinate system, label each line with its
equation, then label the point of intersection with its coordinates.
y
y =x+3
10
(1, 4)
−2
10
−2
x
2x + y = 6
Substitute the point (x, y) = (1, 4) in both equations to see if it checks.
2x + y = 6
2(1) + (4) = 6
6=6
y =x+3
4 = (1) + 3
4=4
Hence, the solution (x, y) = (1, 4) checks.
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208
5. First, determine the x- and y-intercepts of x + 2y = −6.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
x + 2y = −6
x + 2y = −6
x + 2(0) = −6
x = −6
(0) + 2y = −6
2y = −6
y = −3
Plot and label the intercepts, then draw the line x + 2y = −6 through them
and label it with its equation. Next, the line y = −3x − 8 has slope −3 and
y-intercept (0, −8). Plot and label (0, −8), then move 1 unit to the left and 3
units up. Label the resulting line with its equation.
y
y = −3x − 8
2
(−6, 0)
−10
2
x
y
2
−10
2
(0, −3)
x + 2y = −6
Δy = 3
(0, −8)
Δx = −1
−10
−10
Next, place both lines on the same coordinate system, label each line with its
equation, then label the point of intersection with its coordinates.
y = −3x − 8
y
2
−10
2
x
(−2, −2)
x + 2y = −6
−10
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4.1. SOLVING SYSTEMS BY GRAPHING
209
Substitute the point (x, y) = (−2, −2) in both equations to see if it checks.
x + 2y = −6
y = −3x − 8
(−2) + 2(−2) = −6
−6 = −6
−2 = −3(−2) − 8
−2 = −2
Hence, the solution (x, y) = (−2, −2) checks.
7. First, determine the x- and y-intercepts of −x − 3y = 3.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
−x − 3y = 3
−x − 3(0) = 3
−x − 3y = 3
−(0) − 3y = 3
−x = 3
−3y = 3
x = −3
y = −1
Plot and label the intercepts, then draw the line −x − 3y = 3 through them
and label it with its equation.
y
5
(−3, 0)
(0, −1)
−5
5
x
−x − 3y = 3
−5
Secondly, determine the x- and y-intercepts of x − 4y = −4.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
x − 4y = −4
x − 4(0) = −4
x − 4y = −4
(0) − 4y = −4
x = −4
−4y = −4
y=1
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210
Plot and label the intercepts, then draw the line x − 4y = −4 through them
and label it with its equation.
y
5
x − 4y = −4
(0, 1)
(−4, 0)
−5
x
5
−5
Finally, plot both lines on the same coordinate system, label each with its
equation, then label the point of intersection with its approximate coordinates.
y
5
x − 4y = −4
−5
(−3.4, 0.1)
5
x
−x − 3y = 3
−5
Hence, the solution is approximately (x, y) ≈ (−3.4, 0.1).
Check: Remember, our estimate is an approximation so we don’t expect
the solution to check exactly (though sometimes it might). Substitute the
approximation (x, y) ≈ (−3.4, 0.1) in both equations to see how close it checks.
We use a calculator to perform the arithmetic.
−x − 3y = 3
−(−3.4) − 3(0.1) ≈ 3
x − 4y = −4
(−3.4) − 4(0.1) ≈ −4
3.1 ≈ 3
−3.8 ≈ −4
That’s fairly close, suggesting the approximation (x, y) ≈ (−3.4, 0.1) is fairly
close to the proper solution.
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9. First, determine the x- and y-intercepts of −3x + 3y = −9.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
−3x + 3y = −9
−3x + 3(0) = −9
−3x + 3y = −9
−3(0) + 3y = −9
−3x = −9
x=3
3y = −9
y = −3
Plot and label the intercepts, then draw the line −3x + 3y = −9 through
them and label it with its equation.
y
5
−3x + 3y = −9
(3, 0)
−5
5
x
(0, −3)
−5
Secondly, determine the x- and y-intercepts of −3x + 3y = −12.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
−3x + 3y = −12
−3x + 3(0) = −12
−3x + 3y = −12
−3(0) + 3y = −12
−3x = −12
x=4
3y = −12
y = −4
Plot and label the intercepts, then draw the line −3x + 3y = −12 through
them and label it with its equation.
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212
y
5
(4, 0)
−5
x
5
(0, −4)
−5
−3x + 3y = −12
Finally, plot both lines on the same coordinate system and label each with
its equation.
y
5
−3x + 3y = −9
−5
5
x
−5
−3x + 3y = −12
It appears that the lines might be parallel. Let’s put each into slopeintercept form to check this supposition.
Solve −3x + 3y = −9 for y:
Solve −3x + 3y = −12 for y:
−3x + 3y = −9
−3x + 3y = −12
−3x + 3y + 3x = −9 + 3x
3y = −9 + 3x
3y
−9 + 3x
=
3
3
y =x−3
−3x + 3y + 3x = −12 + 3x
3y = −12 + 3x
3y
−12 + 3x
=
3
3
y =x−4
Note that both lines have slope 1. However, the first line has a y-intercept
at (0, −3), while the second line has a y-intercept at (0, −4). Hence, the lines
are distinct and parallel. Therefore, the system has no solutions.
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4.1. SOLVING SYSTEMS BY GRAPHING
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11. First, determine the x- and y-intercepts of 6x − 7y = −42.
To find the y-intercept, let x = 0.
To find the x-intercept, let y = 0.
6x − 7y = −42
6x − 7y = −42
6x − 7(0) = −42
6x = −42
6(0) − 7y = −42
−7y = −42
x = −7
y=6
Plot and label the intercepts, then draw the line 6x − 7y = −42 through
them and label it with its equation. Next, the line y = − 41 x + 4 has slope −1/4
and y-intercept (0, 4). Plot and label (0, 4), then move 4 units to the left and
1 unit up. Label the resulting line with its equation.
y
y
10
10
6x − 7y = −42
(0, 6)
(0, 4)
Δy = 1
Δx = −4
y = − 41 x + 4
(−7, 0)
−10
2
x
−10
2
−2
−2
Next, place both lines on the same coordinate system, label each line with its
equation, then label the point of intersection with its coordinates.
y
10
6x − 7y = −42
(−1.8, 4.5)
y = − 14 x + 4
−10
2
x
−2
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CHAPTER 4. SYSTEMS
214
Hence, the solution is approximately (x, y) ≈ (−1.8, 4.5).
Check: Remember, our estimate is an approximation so we don’t expect
the solution to check exactly (though sometimes it might). Substitute the
approximation (x, y) ≈ (−1.8, 4.5) in both equations to see how close it checks.
We use a calculator to perform the arithmetic.
1
y =− x+4
4
1
4.5 ≈ − (−1.8) + 4
4
4.5 ≈ 4.45
6x − 7y = −42
6(−1.8) − 7(4.5) ≈ −42
−42.3 ≈ −42
That’s fairly close, suggesting the approximation (x, y) ≈ (−1.8, 4.5) is fairly
close to the proper solution.
13. First, determine the x- and y-intercepts of 6x − 7y = −42.
To find the y-intercept, let x = 0.
To find the x-intercept, let y = 0.
6x − 7y = −42
6x − 7y = −42
6x − 7(0) = −42
6x = −42
6(0) − 7y = −42
−7y = −42
x = −7
y=6
Plot and label the intercepts, then draw the line 6x − 7y = −42 through
them and label it with its equation. Next, the line y = − 51 x + 2 has slope −1/5
and y-intercept (0, 2). Plot and label (0, 2), then move 5 units to the left and
1 unit up. Label the resulting line with its equation.
y
y
10
10
6x − 7y = −42
(0, 6)
(0, 2)
Δy = 1
Δx = −5
(−7, 0)
−10
2
−2
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x
y = − 15 x + 2
x
2
−10
−2
4.1. SOLVING SYSTEMS BY GRAPHING
215
Next, place both lines on the same coordinate system, label each line with its
equation, then label the point of intersection with its coordinates.
y
10
6x − 7y = −42
(−3.8, 2.8)
y = − 15 x + 2
x
2
−10
−2
Hence, the solution is approximately (x, y) ≈ (−3.8, 2.8).
Check: Remember, our estimate is an approximation so we don’t expect
the solution to check exactly (though sometimes it might). Substitute the
approximation (x, y) ≈ (−3.8, 2.8) in both equations to see how close it checks.
We use a calculator to perform the arithmetic.
6x − 7y = −42
6(−3.8) − 7(2.8) ≈ −42
−42.4 ≈ −42
1
y = − x+2
5
1
2.8 ≈ − (−3.8) + 2
5
2.8 ≈ 2.76
That’s fairly close, suggesting the approximation (x, y) ≈ (−3.8, 2.8) is fairly
close to the proper solution.
15. First, determine the x- and y-intercepts of 6x + 3y = 12.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
6x + 3y = 12
6x + 3y = 12
6x + 3(0) = 12
6x = 12
6(0) + 3y = 12
3y = 12
x=2
y=4
Plot and label the intercepts, then draw the line 6x + 3y = 12 through them
and label it with its equation.
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216
y
5 (0, 4)
(2, 0)
−5
x
5
−5
6x + 3y = 12
Secondly, determine the x- and y-intercepts of −2x − y = 4.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
−2x − y = 4
−2x − y = 4
−2x − (0) = 4
−2x = 4
−2(0) − y = 4
−y = 4
x = −2
y = −4
Plot and label the intercepts, then draw the line −2x − y = 4 through them
and label it with its equation.
y
−2x − y = 4
5
(−2, 0)
−5
5
x
(0, −4)
−5
Finally, plot both lines on the same coordinate system and label each with
its equation.
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217
y
−2x − y = 4
5
−5
5
−5
x
6x + 3y = 12
It appears that the lines might be parallel. Let’s put each into slopeintercept form to check this supposition.
Solve 6x + 3y = 12 for y:
6x + 3y = 12
6x + 3y − 6x = 12 − 6x
3y = 12 − 6x
12 − 6x
3y
=
3
3
y = −2x + 4
Solve −2x − y = 4 for y:
−2x − y = 4
−2x − y + 2x = 4 + 2x
−y = 4 + 2x
−y
4 + 2x
=
−1
−1
y = −2x − 4
Note that both lines have slope −2. However, the first line has a y-intercept
at (0, 4), while the second line has a y-intercept at (0, −4). Hence, the lines
are distinct and parallel. Therefore, the system has no solutions.
17. First, determine the x- and y-intercepts of 3x + y = 3.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
3x + y = 3
3x + (0) = 3
3x + y = 3
3(0) + y = 3
3x = 3
x=1
y=3
Plot and label the intercepts, then draw the line 3x + y = 3 through them
and label it with its equation.
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218
y
5
(0, 3)
(1, 0)
−5
x
5
−5
3x + y = 3
Secondly, determine the x- and y-intercepts of −2x + 3y = −6.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
−2x + 3y = −6
−2x + 3y = −6
−2x + 3(0) = −6
−2x = −6
−2(0) + 3y = −6
3y = −6
y = −2
x=3
Plot and label the intercepts, then draw the line −2x + 3y = −6 through
them and label it with its equation.
y
5
−2x + 3y = −6
(3, 0)
−5
(0, −2)
5
x
−5
Finally, plot both lines on the same coordinate system, label each with its
equation, then label the point of intersection with its approximate coordinates.
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4.1. SOLVING SYSTEMS BY GRAPHING
219
y
5
−2x + 3y = −6
−5
5
(1.4, −1.1)
−5
x
3x + y = 3
Hence, the solution is approximately (x, y) ≈ (1.4, −1.1).
Check: Remember, our estimate is an approximation so we don’t expect
the solution to check exactly (though sometimes it might). Substitute the
approximation (x, y) ≈ (1.4, −1.1) in both equations to see how close it checks.
We use a calculator to perform the arithmetic.
3x + y = 3
−2x + 3y = −6
3(1.4) + (−1.1) ≈ 3
3.1 ≈ 3
−2(1.4) + 3(−1.1) ≈ −6
−6.1 ≈ −6
That’s fairly close, suggesting the approximation (x, y) ≈ (1.4, −1.1) is fairly
close to the proper solution.
19. Enter the equations y = 34 x + 7 and y = − 13 x + 2 in the Y= menu as
shown in the first image below. Select 6:ZStandard from the ZOOM menu to
sketch the system. Press 2ND CALC to open the Calculate menu (see the
second image below), then select 5:intersect. Press ENTER for “First curve,”
ENTER for “Second curve,” and ENTER for “Guess.” The result is shown in
the third image below.
The calculator reports the solution:
(x, y) ≈ (−4.615385, 3.5384615)
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CHAPTER 4. SYSTEMS
220
Rounding to the nearest tenth, we get (x, y) ≈ (−4.6, 3.5).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
y
10
y = 34 x + 7
y = − 13 x + 2
(−4.6, 3.5)
−10
10
x
−10
21. Enter the equations y = 43 x − 3 and y = − 74 x − 1 in the Y= menu as
shown in the first image below. Select 6:ZStandard from the ZOOM menu to
sketch the system. Press 2ND CALC to open the Calculate menu (see the
second image below), then select 5:intersect. Press ENTER for “First curve,”
ENTER for “Second curve,” and ENTER for “Guess.” The result is shown in
the third image below.
The calculator reports the solution:
(x, y) ≈ (1.05, −1.6)
Rounding to the nearest tenth, we get (x, y) ≈ (1.1, −1.6).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
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4.1. SOLVING SYSTEMS BY GRAPHING
221
y
y = 43 x − 3
10
y = − 47 x − 1
−10
(1.1, −1.6)
10
x
−10
23. Enter the equations y = 16 x + 1 and y = − 37 x + 5 in the Y= menu as
shown in the first image below. Select 6:ZStandard from the ZOOM menu to
sketch the system. Press 2ND CALC to open the Calculate menu (see the
second image below), then select 5:intersect. Press ENTER for “First curve,”
ENTER for “Second curve,” and ENTER for “Guess.” The result is shown in
the third image below.
The calculator reports the solution:
(x, y) ≈ (6.72, 2.12)
Rounding to the nearest tenth, we get (x, y) ≈ (6.7, 2.1).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
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CHAPTER 4. SYSTEMS
222
y
y=
− 37 x
10
+5
y = 16 x + 1
(6.7, 2.1)
−10
10
x
−10
25. First, solve each equation for y, so that we can enter the resulting equations
into the Y= menu.
6x + 16y = 96
6x + 16y − 6x = 96 − 6x
−6x + 13y = −78
−6x + 13y + 6x = −78 + 6x
16y = 96 − 6x
13y = −78 + 6x
96 − 6x
−78 + 6x
16y
13y
=
=
16
16
13
13
3
6
y =6− x
y = −6 + x
8
13
6
Enter the equations y = 6− 38 x and y = −6+ 13
x in the Y= menu as shown
in the first image below. Select 6:ZStandard from the ZOOM menu to sketch
the system. Make the adjustments in the second image shown below so that
the point of intersection of the two lines is visible in the viewing window. Press
2ND CALC to open the Calculate menu, then select 5:intersect. Press ENTER
for “First curve,” ENTER for “Second curve,” and ENTER for “Guess.” The
result is shown in the third image below.
The calculator reports the solution:
(x, y) ≈ (14.344828, 0.62068966)
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223
Rounding to the nearest tenth, we get (x, y) ≈ (14.3, 0.6).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
y
10
6x + 16y = 96
(14.3, 0.6)
−5
20
x
−6x + 13y = −78
−10
27. First, solve each equation for y, so that we can enter the resulting equations
into the Y= menu.
−2x − 11y = 22
−2x − 11y + 2x = 22 + 2x
−11y = 22 + 2x
−11y
22 + 2x
=
−11
−11
2
y = −2 − x
11
8x − 12y = −96
8x − 12y − 8x = −96 − 8x
−12y = −96 − 8x
−96 − 8x
−12y
=
−12
−12
2
y =8+ x
3
2
x and y = 8+ 23 x in the Y= menu as shown
Enter the equations y = −2− 11
in the first image below. Select 6:ZStandard from the ZOOM menu to sketch
the system. Make the adjustments in the second image shown below so that
the point of intersection of the two lines is visible in the viewing window. Press
2ND CALC to open the Calculate menu, then select 5:intersect. Press ENTER
for “First curve,” ENTER for “Second curve,” and ENTER for “Guess.” The
result is shown in the third image below.
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224
The calculator reports the solution:
(x, y) ≈ (−11.78571, 0.14285714)
Rounding to the nearest tenth, we get (x, y) ≈ (−11.8, 0.1).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
y
10
−20
8x − 12y = −96
(−11.8, 0.1)
5
x
−2x − 11y = 22
−10
29. First, solve each equation for y, so that we can enter the resulting equations
into the Y= menu.
−6x + 2y = −12
−12x + 3y = −36
−6x + 2y + 6x = −12 + 6x
2y = −12 + 6x
−12 + 6x
2y
=
2
2
y = −6 + 3x
−12x + 3y + 12x = −36 + 12x
3y = −36 + 12x
3y
−36 + 12x
=
3
3
y = −12 + 4x
Enter the equations y = −6+3x and y = −12+4x in the Y= menu as shown
in the first image below. Select 6:ZStandard from the ZOOM menu to sketch
the system. Make the adjustments in the second image shown below so that
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4.1. SOLVING SYSTEMS BY GRAPHING
225
the point of intersection of the two lines is visible in the viewing window. Press
2ND CALC to open the Calculate menu, then select 5:intersect. Press ENTER
for “First curve,” ENTER for “Second curve,” and ENTER for “Guess.” The
result is shown in the third image below.
The calculator reports the solution:
(x, y) ≈ (6, 12)
Rounding to the nearest tenth, we get (x, y) ≈ (6.0, 12.0).
Using the Calculator Submission Guidelines, report the solution on your
homework paper as follows.
y
−6x + 2y = −12
20
(6.0, 12.0)
−10
10
x
−12x + 3y =
−5−36
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