Appendix A — Answer Key, Chapter 8: Linear Functions and $y = mx + b$

SOL 8.PFA.3 · Covers textbook Chapter 8 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 172 across the chapter. Reasoning and context answers show an acceptable response, not the only wording.

Conventions used throughout:

• Slope-intercept form is $y = mx + b$; $m$ is the slope, $b$ is the y-intercept, $x$ is the independent variable, $y$ is the dependent variable.
• Every listed ordered pair has integer coordinates and every y-intercept is an integer, as the standard requires. Where a slope is fractional, the listed $x$-values are multiples of its denominator on purpose.
• Where a check by substitution is shown, it is there because it is the fastest way to catch a sign error or a swapped $m$ and $b$.
• For items 121–140 and 165–168 any story with the correct structure is acceptable. Grade on four points: what $x$ counts, what $y$ measures, $b$ as the amount present at $x = 0$, and $m$ as the amount added (or removed) per one unit of $x$, with a sign that matches.

Lesson 8.1 — Adding a Constant: Translating $y = mx$

Guided practice

1. The line $y = 3x$ slides straight up $5$ units; the slope stays $3$ and the new y-intercept is $(0, 5)$. Equation: $y = 3x + 5$.
2. It is the graph of $y = -x$ translated down $6$ units, since $y = -x - 6$ is $y = -x + (-6)$ and adding $-6$ lowers every $y$-value by $6$.
3. The parent line is $y = \tfrac{1}{2}x$, translated up $4$ units.
4. $y = 4x - 7$
5. No. A vertical translation moves every point the same distance in the same direction, so the rise and run between any two points are unchanged.
6. $(0, -9)$

Independent practice

7. a) $y = 5x$, up $2$ b) $y = -2x$, down $8$ c) $y = \tfrac{2}{3}x$, up $6$ d) $y = x$, down $1$
8. $y = -3x + 4$
9. $y = \tfrac{1}{4}x - 3$
10. $y = 2x + 3$ is highest, at $(1, 5)$. The other two give $(1, 2)$ and $(1, -2)$.
11. $(2x + 3) - (2x - 4) = 7$. The $x$-terms cancel, so the gap is $7$ units at every $x$-value and does not depend on $x$. The lines are parallel.

Every entry in the third column is $3$ more than the entry beside it, because adding $3$ to $2x$ raises the result by $3$ no matter what $x$ is.

13. No. First, $b = 3 \neq 0$, so the graph passes through $(0, 3)$ and misses the origin, and a proportional relationship must contain $(0,0)$. Second, the ratio $\tfrac{y}{x}$ is not constant: at $x = 1$ it is $\tfrac{5}{1} = 5$ and at $x = 2$ it is $\tfrac{7}{2}$.
14. Both have slope $\tfrac{1}{2}$, so at every $x$-value the first is exactly $(\tfrac{1}{2}x + 3) - (\tfrac{1}{2}x - 2) = 5$ units above the second. A gap of $5$ that never shrinks means the lines can never touch; they are parallel.
15. $y = 2x + 6$. The graph of $y = 2x$ is translated up $6$ units. The rate of filling has not changed, so the slope is still $2$; only the starting height moved, from $(0,0)$ to $(0,6)$.
16. Adding $-5$ changes the $y$-coordinate, not the $x$-coordinate, so the points move down, not sideways. The correct description is the graph of $y = 3x$ translated down $5$ units. Checking one point settles it: $y = 3x$ contains $(0,0)$, and $y = 3x - 5$ contains $(0,-5)$, which is $5$ units below, not $5$ units left.

Exit ticket 8.1

17. $y = 6x + 2$
18. Parent line $y = -x$, translated down $7$ units.
19. $(0, 4)$
20. Slope is the ratio of the rise to the run between two points. A vertical translation adds the same amount to both points' $y$-coordinates and nothing to their $x$-coordinates, so the rise between them and the run between them are both unchanged, and their ratio is unchanged.

Lesson 8.2 — Key Characteristics of a Linear Function

Guided practice

21. $m = 4$, $b = -1$
22. $m = -\tfrac{2}{3}$, $b = 5$
23. The number of miles driven. The cost depends on the miles, not the other way around.
24. It falls. The slope decides it: $m = -1 < 0$, so as $x$ increases $y$ decreases.
25. $(0, 0)$. Since $y = 7x$ is $y = 7x + 0$, this one is proportional.
26. $m = 0$. The graph is a level horizontal line through $(0, 3)$; every input gives $y = 3$.

Independent practice

28. $m = 2$; the y-intercept is $(0, -3)$; three other plotted points are $(1, -1)$, $(2, 1)$, and $(3, 3)$.
29. a) rises b) falls c) level d) rises
30. $y = 3x + 1$ is steeper. Its slope $3 = \tfrac{3}{1}$ climbs $3$ units for a run of $1$, while $\tfrac{1}{2}$ climbs only $1$ unit for a run of $2$ — half a unit per step across.
31. Independent variable: $x$, gigabytes used. Dependent variable: $y$, total cost in dollars. $m = 4$ means the bill grows $4 for each additional gigabyte. $b = 20$ means the bill is $20 when no data has been used — the fixed monthly charge.
32. They must share the point $(0, b)$, where both cross the $y$-axis. They must differ in steepness or direction, since their slopes differ, so they are not parallel and cross only at that shared point.
33. The y-intercept is the output at $x = 0$, and substituting $x = 0$ into $y = mx + b$ gives exactly one value, $y = b$. A function assigns exactly one output to each input, so there is exactly one point on the $y$-axis.
34. $m = -2$, $b = 5$. The negative slope means $y$ decreases as $x$ increases: each increase of $1$ in $x$ lowers $y$ by $2$, so the line falls from left to right.
35. $y = 9x + 12$. $m = 9$ dollars per hour, $b = 12$ dollars. Independent variable: $x$, hours worked. Dependent variable: $y$, total charge in dollars.
36. The equation is not yet in slope-intercept form; the student read the terms in the order they appear. Rewriting by the commutative property, $y = 8 + (-3x) = -3x + 8$. Correct values: $m = -3$ and $b = 8$ — exactly the reverse of the student's answer.

Exit ticket 8.2

37. $m = -\tfrac{1}{3}$, $b = 6$
38. $m = -5$, $b = 12$
39. The distance is the dependent variable, because its value is determined by how long the car has been driving. Time is the independent variable.
40. The slope tells you the rate: how fast the output changes, in what direction, and how steep the line is. The y-intercept tells you only the single starting value at $x = 0$; it says nothing about how the function behaves after that. Two lines can share a y-intercept and behave completely differently.

Lesson 8.3 — Graphing a Linear Function from an Equation

Each answer lists the y-intercept first, then the points reached by stepping with the slope. Any three correct lattice points on the line are acceptable.

Guided practice

41. $(0, -3)$, $(1, -1)$, $(2, 1)$. Plot $b = -3$, then right $1$, up $2$.
42. $(0, 4)$, $(1, 3)$, $(2, 2)$. Slope $-1 = \tfrac{-1}{1}$: right $1$, down $1$.
43. $(0, 1)$, $(2, 2)$, $(4, 3)$. Slope $\tfrac{1}{2}$: right $2$, up $1$.
44. $(0, 4)$, $(2, 1)$, $(4, -2)$. Slope $-\tfrac{3}{2}$: right $2$, down $3$. Check: $-\tfrac{3}{2}(4) + 4 = -2$. ✓
45. $(0, 0)$, $(1, 3)$, $(2, 6)$. Since $b = 0$, this line passes through the origin and is proportional.
46. $(-3, -4)$, $(0, -4)$, $(3, -4)$. Rewrite as $y = 0x - 4$: a level line.

Independent practice

47. $(0, -2)$, $(3, 0)$, $(6, 2)$. Slope $\tfrac{2}{3}$: right $3$, up $2$. (Stepping backwards also gives $(-3, -4)$.)
48. $(0, 5)$, $(1, 3)$, $(2, 1)$
49. $(-4, 2)$, $(0, 3)$, $(4, 4)$. Slope $\tfrac{1}{4}$: right $4$, up $1$; reversing gives left $4$, down $1$.
50. $(-2, 0)$, $(0, -1)$, $(2, -2)$
51. $(0, -6)$, $(1, -1)$, $(2, 4)$
52. Both have slope $1$, so the lines are parallel. $y = x + 2$ passes through $(0,2)$ and $y = x - 3$ through $(0,-3)$, and $(x + 2) - (x - 3) = 5$, so they are $5$ units apart at every $x$-value.
53. Because $b$ is a point you can plot without any computation — it is sitting in the equation — and $m$ then gives you a direction to step in. A table requires a substitution for every row before you can plot anything.
54. $y = -2x + 12$. The graph starts at $(0, 12)$ and falls $2$ for each step right: $(1,10)$, $(2,8)$, $(3,6)$, $(4,4)$, $(5,2)$, $(6,0)$. The point $(6, 0)$ means the candle's height is $0$ after $6$ hours — it has burned out — so only $x$-values from $0$ to $6$ make sense in the story.
55. $y = 2x + 3$. Points: $(0,3)$, $(1,5)$, $(2,7)$, $(3,9)$, $(4,11)$, $(5,13)$, $(6,15)$. A $5$-mile ride costs $13, and the equation confirms it: $2(5) + 3 = 13$.
56. The student swapped the rise and the run. The move right $3$ and down $1$ has $\tfrac{\text{rise}}{\text{run}} = \tfrac{-1}{3}$, so the student graphed slope $-\tfrac{1}{3}$ instead of $-3$. Since $-3 = \tfrac{-3}{1}$, the correct move from $(0,2)$ is right $1$ and down $3$, landing on $(1, -1)$.

Exit ticket 8.3

57. $(0, -4)$, $(1, -1)$, $(2, 2)$
58. $(-3, 4)$, $(0, 2)$, $(3, 0)$
59. $(0, -3)$, $(2, -2)$, $(4, -1)$
60. If $m$ is positive, move up after running to the right; if $m$ is negative, move down. (If $m = 0$, do not move vertically at all — the line is level.) The sign of the rise is the sign of the slope.

Lesson 8.4 — Tables of Values for Linear Functions

Guided practice

Each step of $1$ in $x$ raises $y$ by $4$, matching $m = 4$; the row $x = 0$ shows $b = -1$.

Check a row against $y = 3x - 2$: $3(2) - 2 = 4$. ✓

66. The $y$-values must change by the same amount from row to row, and that constant amount is the slope $m$. Constant rate of change is what makes the function linear.

Independent practice

The constant change in $y$ is $3$, which is $m$.

Check: $-\tfrac{3}{2}(-2) + 4 = 3 + 4 = 7$. ✓

69. Use multiples of $3$, the denominator of $m = \tfrac{2}{3}$, so that $\tfrac{2}{3}x$ is a whole number.

The slope is $0$; the table shows it because $y$ does not change at all as $x$ increases, so the change in $y$ over the change in $x$ is $\tfrac{0}{1} = 0$.

The last column says that after $5$ minutes the tank holds $0$ liters — it is empty. The story stops there, even though the equation would keep going.

74. Not linear. With equal $x$-steps of $1$, the $y$-values change by $2$, then $3$, then $4$. The rate of change is not constant, so no equation $y = mx + b$ fits the table.
75. $y = 10x + 25$.

Continuing: $75 at week $5$ and $85 at week $6$. Check: $10(6) + 25 = 85$. ✓

76. With $m = \tfrac{1}{3}$, the product $\tfrac{1}{3}x$ is only a whole number when $x$ is a multiple of $3$; the chosen values $1, 2, 4, 5$ are not, so each output picked up a fraction. Better choices: $x = -3, 0, 3, 6$, which give $y = 1, 2, 3, 4$.

Exit ticket 8.4

79. Use multiples of $4$, the denominator of $\tfrac{3}{4}$.

80. Find the row where $x = 0$; the $y$-value in that row is $b$, because substituting $x = 0$ into $y = mx + b$ gives $y = b$. If no row shows $x = 0$, find $m$ first and then step back to $x = 0$, or substitute a known point into $y = mx + b$ and solve for $b$.

Lesson 8.5 — Writing $y = mx + b$ from a Graph or a Table

Guided practice

81. $y = -2x + 5$. The line crosses the $y$-axis at $(0,5)$, so $b = 5$; from $(1,3)$ to $(3,-1)$ the rise is $-4$ and the run is $2$, so $m = \tfrac{-4}{2} = -2$.
82. $y = 2x - 3$. From $(0,-3)$ to $(1,-1)$: rise $2$, run $1$, so $m = 2$.
83. $m = 3$, $b = 4$, so $y = 3x + 4$. Check the last column: $3(3) + 4 = 13$. ✓
84. $m = -2$, $b = 9$, so $y = -2x + 9$. Check: $-2(3) + 9 = 3$. ✓
85. $m = \tfrac{3 - (-1)}{2 - 0} = \tfrac{4}{2} = 2$ and $b = -1$, so $y = 2x - 1$.
86. $m = \tfrac{1}{2}$ (the $y$-values rise $1$ for every $2$ in $x$), $b = 5$, so $y = \tfrac{1}{2}x + 5$. Check: $\tfrac{1}{2}(6) + 5 = 8$. ✓

Independent practice

87. a) $m = 3$, $b = 2$: $y = 3x + 2$ b) $m = \tfrac{3}{3} = 1$, $b = -4$: $y = x - 4$ c) $m = \tfrac{-4}{2} = -2$, $b = 6$: $y = -2x + 6$ d) $m = \tfrac{3}{4}$, $b = 0$: $y = \tfrac{3}{4}x$ — proportional, since it contains the origin.
88. $m = 3$, $b = -4$, so $y = 3x - 4$. Check: $3(2) - 4 = 2$. ✓
89. The $x$-values step by $2$ while $y$ drops by $3$, so $m = -\tfrac{3}{2}$; the row $x = 0$ gives $b = 6$. Equation: $y = -\tfrac{3}{2}x + 6$. Check: $-\tfrac{3}{2}(4) + 6 = 0$. ✓
90. $m = 2$. No row shows $x = 0$, so step back from $(1, 1)$: going left $1$ means going down $2$, giving $y = -1$ at $x = 0$, so $b = -1$. Equation: $y = 2x - 1$. (By substitution: $1 = 2(1) + b$ gives $b = -1$.) Check: $2(4) - 1 = 7$. ✓
91. $m = \tfrac{-1}{2} = -\tfrac{1}{2}$. Substituting $(2,-1)$: $-1 = -\tfrac{1}{2}(2) + b = -1 + b$, so $b = 0$. Equation: $y = -\tfrac{1}{2}x$. It is proportional, because $b = 0$ and the graph passes through the origin.
92. $y = -3x + 7$. The given point lies on the $y$-axis, so it is the y-intercept.
93. $y = 2x + 8$, where $x$ is shirts. For $15$ shirts, $2(15) + 8 = 38$ dollars.
94. $y = -3x + 24$. The tank is empty when $y = 0$: $-3x + 24 = 0$ gives $x = 8$ minutes.
95. First student: $m = \tfrac{6 - 4}{2 - 0} = \tfrac{2}{2} = 1$. Second student: $m = \tfrac{10 - 6}{6 - 2} = \tfrac{4}{4} = 1$. Both read $b = 4$ at the $y$-axis, so both write $y = x + 4$. It had to happen because a linear function has a constant rate of change: the ratio of rise to run is the same between any two of its points, so no choice of slope triangle can change the answer.
96. The student put the two numbers in the wrong slots. The value at $x = 0$ is $5$, so $b = 5$; the constant change per step is $3$, so $m = 3$. Correct equation: $y = 3x + 5$. Testing the student's version at $x = 2$ gives $5(2) + 3 = 13$, but the table says $11$.

Exit ticket 8.5

97. $y = 4x - 2$
98. The $x$-values step by $3$ while $y$ rises by $2$, so $m = \tfrac{2}{3}$; $b = -1$. Equation: $y = \tfrac{2}{3}x - 1$. Check: $\tfrac{2}{3}(6) - 1 = 3$. ✓
99. $m = \tfrac{1 - 5}{1 - 0} = -4$ and $b = 5$, so $y = -4x + 5$.
100. Find $m$ first, as the change in $y$ divided by the change in $x$. Then either step backwards to $x = 0$, undoing one $m$ for each step left, or substitute any known ordered pair into $y = mx + b$ and solve for $b$. Both routes give the same value.

Lesson 8.6 — Linear Functions in Context

Guided practice

101. $b = 5$. It is the $5 unlock charge — the cost when $x = 0$, before any riding.
102. $m = 3$. It is the rate: the cost grows $3 for each additional hour.
103. $3(7) + 5 = 26$ dollars.
104. After $5$ minutes the tank holds $0$ liters, so the tank is empty at $5$ minutes. It is also where the story ends, since the tank cannot hold a negative amount.
105. $y = 9x + 12$
106. Independent variable: $x$, the number of hours worked. Dependent variable: $y$, the total charge in dollars.

Independent practice

107. a) $y = 12x + 20$ b)

 c) $12(10) + 20 = 140$ dollars.

108. a) $y = -6x + 30$ b)

 c) At $x = 5$ seconds, since that is where $y = 0$.

109. $y = 2x + 4$. Points: $(0,4)$, $(1,6)$, $(2,8)$, $(3,10)$, $(4,12)$, $(5,14)$, $(6,16)$. After $6$ weeks there are $16$ members.

The $x$-column, the number of miles, holds the independent variable.

111. The graph passes through the lattice point $(4, 17)$, so $4$ hours costs $17. The equation agrees: $3(4) + 5 = 17$. ✓
112. In the story $x$ counts hours of riding, and there is no such thing as riding for $-2$ hours. The equation is happy to compute $3(-2) + 5 = -1$, but a cost of $-\$1$ describes nothing in the situation. A context restricts which inputs are meaningful even though the equation does not.
113. The second plan, $y = 5x + 25$, is always more expensive — by $15, at every value of $x$, since $(5x + 25) - (5x + 10) = 15$. The graphs are parallel lines with the same slope $5$, the second sitting $15$ units above the first: exactly the vertical translation of Lesson 8.1.
114. $y = 5x + 30$, where $x$ is gigabytes.

For $9$ gigabytes: $5(9) + 30 = 75$ dollars.

115. $y = -3x + 15$. After $4$ hours: $-3(4) + 15 = 3$ centimeters. It burns out when $y = 0$: $-3x + 15 = 0$ gives $x = 5$ hours.
116. Correct equation: $y = 4x + 6$. The test is the word "per": $4 per package is charged once for each package, so it multiplies the number of packages and is $m$; the $6 handling fee is charged once no matter what, so it is $b$. The student's version, $y = 6x + 4$, charges $6 per package and a $4 fee — it happens to give $10 for one package, but for two it gives $16 instead of the correct $4(2) + 6 = 14$.

Exit ticket 8.6

117. $y = 4x + 10$

119. The y-intercept $(0, 10)$ means the pool already held $10$ inches of water before the hose was turned on — the depth at time $0$.
120. Ask which number is attached to the word "per," or to a phrase like "each hour" or "every mile." That number is the rate and multiplies the independent variable, so it is $m$. The number charged or measured only once, before any counting begins, is $b$. A quick check: substitute $x = 0$ and see whether the result matches the one-time amount.

Lesson 8.7 — Creating a Context for a Linear Function

Any story with the correct structure is acceptable. The responses below are models, and each names $x$, $y$, $b$, and $m$ and includes a check.

Guided practice

121. $x$ counts dogs walked; $y$ measures total dollars a student has. She starts with $20 ($b = 20$) and earns $5 for each dog she walks ($m = 5$). Check: at $x = 3$, $y = 5(3) + 20 = 35$, meaning $35 after walking three dogs.
122. A phone battery is at $16$ percent ($b = 16$) and loses $2$ percentage points for each minute of video played ($m = -2$); $x$ is minutes and $y$ is percent remaining. The quantity decreases, matching the negative slope. Check: at $x = 3$, $y = 10$ percent.
123. Any one-time starting amount: $50 already in a savings account, $50$ liters already in a tank, a $50$-dollar registration fee, $50$ members already in a club. It must be the amount present before the counting variable starts.
124. It must mean something is being removed or used up at $3$ units per step — spending $3 a week, using $3$ cups of flour per batch, losing $3$ meters of altitude per second — starting from $12$.
125. A puppy weighs $6$ pounds today ($b = 6$) and gains $2$ pounds each week ($m = 2$); $x$ is weeks from now and $y$ is weight in pounds. Check: at $x = 4$, $y = 2(4) + 6 = 14$, so the puppy weighs $14$ pounds after four weeks. That sentence makes sense, so the context fits.
126. Because $b$ is the value of $y$ when $x = 0$, and substituting $x = 0$ into $y = 4x + 7$ gives $y = 7$. Whatever $y$ measures in the story, there must be $7$ of it before the counting starts.

Independent practice

127. a) A club has $10$ members and adds $3$ per week; $x$ is weeks, $y$ is members. b) A $40$-page notebook loses $5$ pages a day as sheets are torn out; $x$ is days, $y$ is pages left. Negative slope, so the quantity shrinks. c) A seedling is $4$ centimeters tall and grows $1$ centimeter every $2$ weeks; $x$ is weeks, $y$ is height in centimeters. The fractional slope is stated as one unit per two steps. d) A worker earns $8 per hour with no bonus; $x$ is hours, $y$ is dollars earned. Since $b = 0$, the pay is $0 at $x = 0$ and the relationship is proportional.
128. $m = 3$ and $b = 12$, so $y = 3x + 12$. Context: a photographer charges $12 to book plus $3 per printed photo; $x$ is photos and $y$ is total dollars. Check: at $x = 3$, $y = 21$, matching the table.
129. $y = -3x + 9$. Context: a cook has $9$ cups of flour and uses $3$ cups per batch of biscuits; $x$ is batches and $y$ is cups left. The $x$-intercept is $(3, 0)$: after three batches the flour is gone, so the story stops at $x = 3$.
130. Context: a jar holds $2$ marbles and $6$ more are added each day; $x$ is days and $y$ is marbles. At $x = 5$, $y = 6(5) + 2 = 32$, meaning the jar holds $32$ marbles after five days.
131. Money story: a summer camp costs $100 to register plus $25 per day; $x$ is days and $y$ is total dollars. Non-money story: a reservoir holds $100$ liters and a pump adds $25$ liters per minute; $x$ is minutes and $y$ is liters. Both have $b = 100$ and $m = 25$.
132. Because $m = -4$ is negative, so the quantity decreases by $4$ for every step of $x$, and it begins at only $20$. Solving $-4x + 20 = 0$ gives $x = 5$, so it runs out at $x = 5$ and the story cannot continue past that point.
133. Something would have to already be counted before the driving starts — $5$ miles already on the trip odometer, for instance. Then $y = 60x + 5$ gives the total distance recorded after $x$ hours, with $b = 5$ as the miles present at $x = 0$. Without such a starting amount, $b$ must be $0$ and the equation is $y = 60x$.
134. Context: a puppy weighs $6$ pounds and gains $2$ pounds per week; $x$ is weeks and $y$ is pounds. The point $(4, 14)$ means the puppy weighs $14$ pounds four weeks from now. Check: $2(4) + 6 = 14$. ✓
135. Context: a $10$-centimeter candle burns $1$ centimeter every $2$ days; $x$ is days and $y$ is height in centimeters. At $x = 20$, $y = -\tfrac{1}{2}(20) + 10 = 0$, so the candle is completely gone after $20$ days, and the story ends there.
136. The student made $15 the monthly rate, but in $y = 7x + 15$ the number multiplying $x$ — the per-month amount — is $7$, and the one-time amount is $15$. Corrected context: a gym charges $15 to join plus $7 each month; $x$ is months and $y$ is total dollars paid. Check: after two months, $7(2) + 15 = 29$ dollars, which is $15 plus two $7 payments. ✓

Exit ticket 8.7

137. $x$ counts hours babysat; $y$ measures total dollars earned. A sitter is given a $5 travel payment ($b = 5$) plus $9 per hour ($m = 9$).
138. A $30$-liter drum loses $6$ liters per hour to a leak; $x$ is hours and $y$ is liters remaining. The $x$-intercept is $(5, 0)$, which means the drum is empty after $5$ hours.
139. It means that after $3$ hours of babysitting the sitter has earned $32. Check: $9(3) + 5 = 32$. ✓
140. First: what does $x$ count, and what does $y$ measure? Second: what does $b$ represent as the amount of $y$ present when $x = 0$, and what does $m$ represent as the amount added or removed for each one more $x$ — including whether its sign calls for something growing or something shrinking.

Chapter 8 Review

Part A — Adding a constant translates $y = mx$ (8.PFA.3a)

141. It is the graph of $y = 5x$ translated down $3$ units. The slope stays $5$; the y-intercept moves from $(0,0)$ to $(0,-3)$.
142. $y = -2x + 6$
143. Both have slope $\tfrac{1}{2}$, and $(\tfrac{1}{2}x + 3) - (\tfrac{1}{2}x - 2) = 5$ for every $x$. A constant vertical gap of $5$ units means the lines never meet, which is what parallel means.