Algebra 1 Workbook — Chapter 6: Writing Equations of Lines

SOL A.F.1 (d, e) · Companion to Textbook Chapter 6

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 123.

PAGE 1 — Chapter opener

Chapter 6 · Writing Equations of Lines

Standard A.F.1 (d, e)

In this chapter you will:

• Write a line's equation from a slope and one point with integer coordinates
• Write a line's equation from two points with integer coordinates
• Write a line's equation from its graph, with or without a visible $y$-intercept
• Write a horizontal line as $y = c$ and a vertical line as $x = a$
• Write a line parallel or perpendicular to a given line through a given point
• Check every equation twice — by substitution and by graphing

Words to know: point-slope form · slope-intercept form · slope · $y$-intercept · lattice point · horizontal line · vertical line · parallel · perpendicular · negative reciprocal · model

The one form to memorize: $y - y_1 = m(x - x_1)$. Put the slope in the $m$ slot and the point in the $x_1$ and $y_1$ slots, then solve for $y$.

Convention: parallel means equal slopes. Perpendicular means the slopes multiply to $-1$. A vertical line $x = a$ has no slope and is not a function.

PAGE 2 — Point-slope form

6.1 From a Slope and a Point

FIGURE: fig1-slope-and-a-point.png (full width)

Fill in the form.

$y - \underline{\hspace{2cm}} = \underline{\hspace{2cm}}\,(x - \underline{\hspace{2cm}})$ — the slope goes in the ______ slot, the point goes in the ______ and ______ slots.

1. Given slope: ______ Given point: ____________

2. Point-slope form: _______________________

   Why does $y - (-1)$ become $y + 1$? _______________________________________

3. Solve for $y$: _______________________ $y$-intercept revealed: ____________

4. The dashed triangle shows run ______ and rise ______, so $m =$ ______

PAGE 3 — Practice · slope and a point

Practice · One Point Is Enough

5. Slope $-4$ through $(1, 5)$

   Point-slope: _______________________ Slope-intercept: _______________________

6. Slope $\tfrac12$ through $(-6, 2)$: _______________________

   Check by substitution: _______________________________________

7. Write each in point-slope form, then slope-intercept form.

PAGE 4 — Practice · special cases and reasoning

Practice · When a Step Disappears

8. Slope $0$ through $(5, -2)$: _______________ What kind of line? ____________________

9. Slope $-5$ through $(0, 7)$: _______________

   Why was point-slope form not needed? _______________________________________

10. Explain. Why does point-slope form work with any point, while $y = mx + b$ needs the point $(0, b)$?

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11. Slope $4$ through $(-1, -6)$: _______________ Check: _______________

12. Slope $-\tfrac35$ through $(10, -4)$: _______________ Check: _______________

PAGE 5 — Errors, models, and technology

Apply It · Slope and a Point

12. Find the error. Given slope $3$ through $(2, -1)$, a student writes $y - 1 = 3(x - 2)$ and reports $y = 3x - 5$.

    What went wrong? _______________________________________________

    Correct equation: _______________ The check that catches it: _______________

13. Apply it. A pool drains at $12$ gallons per minute; after $4$ minutes it holds $200$ gallons.

    $W(t) =$ _______________ Slope means _______________________________________

    $y$-intercept means _______________________________________

14. Apply it. A gym charges $35 per month; after $6$ months a member has paid $260.

    $C(m) =$ _______________ The $y$-intercept is _______________

    Why can it not be a monthly charge? _______________________________________

15. Technology. Graph your answer to 7a and look for $(3, 1)$.

    On the line? ______ If it were not, what would you do? _______________________

PAGE 6 — Exit ticket 6.1

Exit Ticket · Lesson 6.1

Name: ________________________ Date: ____________

17. Slope $6$ through $(2, 5)$: _______________________

18. Slope $-\tfrac12$ through $(-4, 1)$: _______________________

FIGURE: fig1-slope-and-a-point.png (half width)

19. Point-slope: _______________________ Slope-intercept: _______________________

    Point each form displays: ____________ and ____________

20. Explain. Why substitute the given point back in, every time?

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PAGE 7 — Two points

6.2 From Two Points

FIGURE: fig2-two-points-integer-coordinates.png (full width)

The four steps. Find the ____________ · use ____________ point in point-slope form · solve for ______ · check with the point you ____________ use.

21. The two marked points: ____________ and ____________

    $m = \dfrac{\underline{\hspace{2cm}}}{\underline{\hspace{2cm}}} =$ ______

22. Point-slope: _______________________ Slope-intercept: _______________________

23. Check with the other point: _______________________________________

PAGE 8 — Practice · two points

Practice · Two Points, One Line

24. Through $(0, 5)$ and $(4, 13)$: _______________ Why no substitution step? ____________

25. Through $(-2, 7)$ and $(3, -3)$: _______________

26. Write each equation in slope-intercept form.

28. Through $(-3, 6)$ and $(3, -2)$: _______________ Both checks: _______________

PAGE 9 — When the slope formula surprises you

Practice · Zero on Top, Zero on the Bottom

29. Through $(2, 7)$ and $(5, 7)$

    Slope formula gives ______ Equation: ____________ Kind of line: ____________

30. Through $(-4, 1)$ and $(-4, 6)$

    Slope formula gives ______ Equation: ____________ A function? ______

31. Explain. Write the line through $(-2, 7)$ and $(3, -3)$ twice, once from each point.

    From $(-2,7)$: _______________ From $(3,-3)$: _______________

    Why did that have to happen? _______________________________________

32. Find the error. For $(1,2)$ and $(3,8)$ a student computes $\dfrac{3-1}{8-2} = \dfrac13$.

    What went wrong? _______________________ Correct slope: ______ Equation: ____________

PAGE 10 — Two measurements make a model

Apply It · Two Points in the Real World

FIGURE: fig6-context-two-points-plumber.png (full width)

31. Apply it. From the two billed jobs: $C(h) =$ _______________

    Slope means _______________________________________ (units: ____________)

    $y$-intercept means _______________________________________

32. Apply it. A candle is $10$ in. tall after $1$ hour and $4$ in. tall after $4$ hours.

    $H(t) =$ _______________ Slope means ____________________________

    $y$-intercept means ____________________________

33. Apply it. A tree is $5$ ft tall at age $3$ and $13$ ft tall at age $7$.

    Model: _______________ Slope means ____________________________

    Why is the $y$-intercept not believable? _______________________________________

PAGE 11 — Checking what you wrote

Two Checks, Every Time

FIGURE: fig10-verify-a-written-equation.png (full width)

26. The two checks shown are ____________________ and ____________________

    If they disagreed, trust ____________________ because _______________________

27. Technology. Graph your answer to 27d and look for both points.

    Both on the line? ______

    A miss at only one point points to an error in the ____________________

    A miss at both points points to an error in the ____________________

PAGE 12 — Exit ticket 6.2

Exit Ticket · Lesson 6.2

Name: ________________________ Date: ____________

37. Through $(0, -3)$ and $(2, 1)$: _______________________

38. Through $(-5, 2)$ and $(1, -4)$: _______________________

39. Through $(4, -1)$ and $(4, 3)$: ____________ Slope: ____________

40. Apply it. A $3$-mile ride costs $11 and a $7$-mile ride costs $19.

    $C(d) =$ _______________ Slope means ____________________________

    $y$-intercept means ____________________________

PAGE 13 — Reading a line off a grid

6.3 From a Graph

FIGURE: fig3-line-from-a-graph.png (full width)

The three steps. Read ______ at the $y$-axis · count ______ between two lattice points · write $y = mx + b$.

41. The line crosses the $y$-axis at ____________, so $b =$ ______

42. The triangle runs between ____________ and ____________, so $m =$ ______

43. Equation: _______________________

44. Check with the marked point $(6, 2)$: _______________________________________

PAGE 14 — When there is no intercept to read

The Harder Graph

FIGURE: fig4-graph-with-no-visible-intercept.png (full width)

44. Why does step 1 fail here? _______________________________________________

45. The two marked lattice points: ____________ and ____________ $m =$ ______

46. Point-slope: _______________________ Slope-intercept: _______________________

    $y$-intercept predicted: ____________ Why it was not visible: ____________________

47. Check with the marked point $(6, 5)$: _______________________________________

PAGE 15 — Practice · from a graph

Practice · Graph to Equation

47. Explain. The three steps, in your own words, and which one fails on the second figure.

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48. Crosses $(0, 4)$, passes $(3, 6)$: _______________________

49. Crosses $(0, -1)$, passes $(2, -7)$: _______________________

50. Passes $(-4, 5)$ and $(2, 2)$, no visible intercept: _______________________

51. Crosses $(0, 2)$, triangle shows run $1$ and rise $-4$: _______________________

PAGE 16 — Reasoning, errors, and technology

Reading Carefully

54. Explain. Why count between lattice points instead of points inside a square?

    ---

55. Find the error. From the first figure of this lesson a student writes $y = \tfrac32 x - 2$.

    What went wrong? _______________________ Correct: _______________

    A point that proves the student wrong: ____________

56. Technology. Graph $y = \tfrac23 x - 2$ and compare it with the figure.

    Feature 1 to check: ____________________ Feature 2: ____________________

PAGE 17 — Exit ticket 6.3

Exit Ticket · Lesson 6.3

Name: ________________________ Date: ____________

57. Crosses $(0, 5)$, passes $(4, 3)$: _______________________

58. Passes $(-6, -5)$ and $(2, -1)$: _______________________

FIGURE: fig4-graph-with-no-visible-intercept.png (half width)

59. Equation: _______________________ $y$-intercept: ____________

60. Explain. When can you read $b$ off a graph, and what do you do when you cannot?

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PAGE 18 — The two special lines

6.4 Horizontal and Vertical Lines

FIGURE: fig5-horizontal-and-vertical-lines.png (full width)

Complete the frame. A horizontal line is written ____________, has slope ______, and ______ a function. A vertical line is written ____________, has ____________, and ______ a function.

61. The blue points share their ______-coordinate, so the equation is ____________

62. The red points share their ______-coordinate, so the equation is ____________

63. Which line is a function? ____________ What goes wrong for the other?

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PAGE 19 — Practice · $y = c$ and $x = a$

Practice · Naming One Coordinate

64. Horizontal line through $(7, -6)$: ____________

65. Vertical line through $(7, -6)$: ____________

66. Explain. Why can $x = 7$ not be written as $y = mx + b$?

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67. Write each equation in the second column.

68. Give each line's slope in the third column, writing "no slope" where it applies.

69. Say in the fourth column whether each line is a function.

The test you used for item 69: _______________________________________

70. Horizontal through $(0, -7)$: ____________ crosses the ______-axis

    Vertical through $(5, 0)$: ____________ crosses the ______-axis

PAGE 20 — Apply it, errors, technology

Apply It · Lines That Do Not Change

71. Apply it. A thermostat holds a room at $68$ degrees all day.

    $T(h) =$ ____________ Slope: ______ (units: ____________)

    What the graph looks like: ____________________________

72. Apply it. A garage wall runs north–south, $12$ feet east of a map's origin.

    Equation: ____________ Why is it not a function of $x$? _______________________

73. Find the error. Asked for the horizontal line through $(4, -1)$, a student writes $x = -1$.

    Mistake 1: ____________________ Mistake 2: ____________________

    Correct: ____________

74. Technology. Enter $y = -3$, then $x = 4$.

    What happens each time? _______________________________________

    What it tells you: _______________________________________

PAGE 21 — Exit ticket 6.4

Exit Ticket · Lesson 6.4

Name: ________________________ Date: ____________

75. Horizontal through $(-2, 5)$: ____________

76. Vertical through $(-2, 5)$: ____________

77. Through $(0, -4)$ and $(9, -4)$: ____________ Slope: ______

78. Explain. $x = a$ versus $y = c$ — which is which, what slope each has, which is a function.

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PAGE 22 — Parallel lines

6.5 Parallel and Perpendicular Lines

FIGURE: fig7-parallel-through-a-point.png (full width)

Two rules. Parallel lines have ____________ slopes. Perpendicular slopes multiply to ______.

79. Given line's slope: ______ Parallel line's slope: ______

    Why must they be equal? _______________________________________

80. Equation of the parallel line through $(1, 4)$: _______________ Check: ____________

PAGE 23 — Perpendicular lines

Turning a Quarter Turn

FIGURE: fig8-perpendicular-through-a-point.png (full width)

81. Blue line's slope: ______ Product with the given slope: ______

82. Equation of the perpendicular line through $(4, 1)$: _______________ Check: ____________

FIGURE: fig9-negative-reciprocal-slopes.png (full width)

83. The two slopes: ______ and ______ Their product: ______

84. Explain. How do the two triangles show "flip the fraction and change the sign"?

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PAGE 24 — Practice · parallel and perpendicular

Practice · Copy It or Flip It

85. Parallel to $y = 3x - 5$ through $(2, 4)$: _______________

86. Parallel to $y = -\tfrac34 x + 1$ through $(8, -2)$: _______________

87. Perpendicular to $y = 4x + 7$ through $(8, 3)$: _______________

88. Perpendicular to $y = -\tfrac23 x + 5$ through $(-4, 0)$: _______________

89. Parallel to $2x + y = 6$ through $(-1, 3)$

    Converted form: _______________ Answer: _______________

90. Perpendicular to $x - 3y = 9$ through $(2, -5)$

    Converted form: _______________ Answer: _______________

PAGE 25 — The pair with no slope

Practice · Vertical Meets Horizontal

91. Parallel to $y = 2$ through $(5, -8)$: ____________

    Perpendicular to $y = 2$ through $(5, -8)$: ____________

92. Parallel to $x = -3$ through $(7, 1)$: ____________

    Perpendicular to $x = -3$ through $(7, 1)$: ____________

93. Are $y = 5x - 2$ and $y = -5x + 2$ perpendicular? ______ Product of slopes: ______

    Explain: _______________________________________________

94. Find the error. For a line perpendicular to $y = \tfrac25 x + 1$, a student uses $-\tfrac25$.

    What was done, and not done? _______________________________________

    Correct slope: ______ Product check: ______

PAGE 26 — Apply it and verify

Apply It · Streets on a Map

94. Apply it. Main Street follows $y = \tfrac12 x + 3$. The fire station is at $(6, 1)$.

    New street, parallel: _______________ Check: ____________

    Access road, perpendicular: _______________ Check: ____________

95. Technology. Graph your answer to 88 with the given line, on a square window.

    What you see: _______________________________________

    Why a wide window can make a correct answer look wrong: _______________________

PAGE 27 — Exit ticket 6.5

Exit Ticket · Lesson 6.5

Name: ________________________ Date: ____________

97. Parallel to $y = -4x + 9$ through $(1, -3)$: _______________

98. Perpendicular to $y = \tfrac12 x - 6$ through $(3, 4)$: _______________

99. Parallel to $x = 8$ through $(-2, 6)$: ____________

100. Explain. State both rules, and name the perpendicular pair the product rule cannot handle.

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PAGE 28 — Chapter 6 review · slope and a point

Chapter 6 Review

Part A · From a slope and a point

101. Slope $-2$ through $(4, 3)$: _______________

102. Slope $\tfrac35$ through $(-5, 2)$: _______________

103. Apply it. A drone climbs $8$ m per second and is $50$ m up after $4$ seconds.

     $h(t) =$ _______________ Slope means ____________________________

     $y$-intercept means ____________________________

104. Find the error. Given slope $-3$ through $(-2, 5)$, a student writes $y - 5 = -3(x - 2)$.

     What went wrong? _______________________ Correct: _______________ Check: ____________

PAGE 29 — Chapter 6 review · two points

Chapter 6 Review (continued)

Part B · From two points

105. Through $(-6, 1)$ and $(2, 5)$: _______________

106. Through $(3, -4)$ and $(-3, 8)$: _______________

FIGURE: fig6-context-two-points-plumber.png (half width)

107. $C(h) =$ _______________ Slope means ____________________________

     $y$-intercept means ____________________________

108. Apply it. $100$ flyers cost $45; $300$ flyers cost $95.

     $C(n) =$ _______________ Slope means ____________________________

     $y$-intercept means ____________________________

PAGE 30 — Chapter 6 review · graphs and special lines

Chapter 6 Review (continued)

Part C · From a graph, and the two special lines

FIGURE: fig3-line-from-a-graph.png (half width)

109. Equation: _______________ Where the $b$ came from: ____________________

     Where the $m$ came from: ____________________

FIGURE: fig4-graph-with-no-visible-intercept.png (half width)

110. Equation: _______________ Why $b$ could not be read: ____________________

FIGURE: fig5-horizontal-and-vertical-lines.png (half width)

111. Horizontal: ____________ slope ______ Vertical: ____________ slope ____________

     Which is a function? ____________

112. Horizontal through $(-9, 2)$: ____________ Vertical through $(-9, 2)$: ____________

113. Find the error. A line crosses $(0,3)$ with a triangle of rise $1$ over run $2$; a student writes $y = 2x + 3$.

     What went wrong? _______________________ Correct: _______________

PAGE 31 — Chapter 6 review · parallel and perpendicular

Chapter 6 Review (continued)

Part D · Parallel and perpendicular

114. Parallel to $y = 2x - 7$ through $(-4, 2)$: _______________

115. Perpendicular to $y = 2x - 7$ through $(-4, 2)$: _______________

116. Parallel to $3x + 2y = 8$ through $(2, -1)$

     Converted form: _______________ Answer: _______________

117. Perpendicular to $y = -5x + 1$ through $(10, -4)$: _______________

118. Perpendicular to $y = -1$ through $(3, -1)$: ____________

     Why no product rule? _______________________________________

PAGE 32 — Chapter 6 review · mixed application

Chapter 6 Review (continued)

Part E · Application and verification

119. Apply it. After $3$ months a customer has paid $95; after $8$ months, $220.

     $C(m) =$ _______________ Slope means ____________________________

     $y$-intercept means ____________________________ Paid after $1$ month: ____________

120. Apply it. First plumber: $C(h) = 50h + 70$. Second: $40 fee plus $60 per hour.

     Second plumber's equation: _______________ Same charge at ______ hours

     Cheaper for a $5$-hour job: ____________________

121. Technology. Through $(-2, -3)$ and $(4, 9)$: _______________

     Both substitution checks: _______________________________________

     If graph and substitution disagreed: _______________________________________

122. Explain. Name the two questions behind a graph, two points, and a slope-with-a-point.

     ---

123. Apply it. A bike path follows $y = -\tfrac34 x + 6$.

     Footpath, perpendicular through $(3, 1)$: _______________ Check: ____________

     Service road, parallel through $(8, 2)$: _______________ Check: ____________

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Item order: a few items are placed by figure rather than by number, so that every item on a page can be answered from the figure on that page. Page 4 runs 8–11 then 16; page 9 runs 29, 30, 35, 36; page 11 runs 26 then 34; page 13 ends with 51 and page 14 ends with 52. The numbers still match the textbook exactly, and the answer key is in numerical order.
• Figure widths: fig1, fig2, fig3, fig4, fig5, fig6, fig7, fig8, fig9, and fig10 are full width on first appearance. fig1, fig3, fig4, fig5, and fig6 repeat later at half width; repeat the image rather than asking students to flip back.
• The two graph figures on pages 13 and 14 are the load-bearing pair of Lesson 6.3 — the whole point is that one shows a $y$-axis crossing and the other does not. Keep them on facing or consecutive pages, and do not crop either window, since cropping fig4 would destroy the very feature it exists to show.
• Blank work space: items 10, 20, 23, 26, 35, 44, 47, 54, 60, 63, 66, 78, 79, 84, 93, 100, 118, and 122 are written explanations — give each at least four ruled lines, not a single blank. Items 13, 14, 31, 32, 33, 40, 71, 72, 94, 103, 107, 108, 119, 120, and 123 are applications with two or more parts; leave two lines per part.
• Two blanks per item, deliberately. Most items in this chapter ask for point-slope form and then slope-intercept form. Set two separate blanks, not one, so a student cannot skip the intermediate step — that step is where the sign errors are caught.
• Fractions: items 6, 7c, 7d, 16, 27b, 28, 43, 45, 46, 50, 53, 57, 58, 59, 82, 83, 86, 87, 88, 90, 96, 98, 102, 109, 110, 113, 115, 116, 117, and 123 involve fractional slopes. Set them as stacked fractions, not slashes, so that $-\tfrac34$ never reads as $-3/4$ ambiguously across a line break.
• Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
• Currency: dollar amounts appear on pages 5, 10, 12, 26, 29, and 32. Set them as plain text ($35, $260, $11, $19, $45, $95, $40, $220), not inside a math box.
• Check marks: every "Check:" blank expects a substitution written out, not the word "yes." Make those blanks wide enough for a full line of arithmetic.