Grade 7 Workbook — Chapter 10: Two-Step Linear Equations

SOL 7.PFA.3 · Companion to Textbook Chapter 10

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/. Item numbers match the textbook exactly, and the problems are the same problems with the same numbers.

PAGE 1 — Chapter opener

Chapter 10 · Two-Step Linear Equations

Standard 7.PFA.3

In this chapter you will:

• Model two-step equations with a balance scale, algebra tiles, and colored chips
• Solve two-step equations by undoing the operations in reverse order
• Name the property of equality that makes each move legal
• Confirm every solution by substituting it back into the original equation
• Write an equation for a situation, and write a situation for an equation
• Solve problems in context and state the answer with units

Words to know: equation · solution · two-step linear equation · coefficient · constant term · substitution · properties of equality · additive inverse property · multiplicative identity property · reciprocal · algebra tiles · zero pair · colored chips · bar model

Coefficients and numeric terms in this chapter are rational numbers: integers, fractions, and decimals.

PAGE 2 — What makes an equation two-step

10.1 What It Means to Solve an Equation

FIGURE: fig5-undo-in-reverse.png (full width)

Fill in the blanks.

A two-step linear equation has the form $ax + b = c$, where $a$ is the ______________ and $b$ is the ______________ term.

Going forward, the variable is ______________ by $a$ and then $b$ is ______________.

Going backward, you undo in the ______________ order.

Complete the table.

PAGE 3 — Checking by substitution

Substitution Is the Habit

FIGURE: fig6-solution-on-number-line.png (full width)

Always substitute into the ORIGINAL equation. Simplify each side separately. Then compare.

Guided practice.

1. Is $x = 3$ a solution of $4x - 5 = 7$? Show the substitution.

   $4(\underline{\quad}) - 5 = \underline{\quad} - 5 = \underline{\quad}$ Right side: ______ Solution? ______

2. Which value from $-2$, $1$, and $4$ is the solution of $2k - 3 = 5$? ______

   Show all three tests: _______________________________________________

3. Name the coefficient and the constant term in $7x - 9 = 12$.

   Coefficient: ______ Constant term: ______ Undo first: _______________

4. Show the check that $x = -5$ is the solution of $-2x + 1 = 11$.

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PAGE 4 — Independent practice

Testing Solutions

5. Decide whether each value is a solution. Circle YES or NO and show the substitution.

6. Which value from $-3$, $3$, and $12$ is the solution of $9 - 2x = 15$? ______

7. Name the coefficient and the constant term.

   a) $5x + 12 = 2$ → coefficient ______, constant ______

   b) $\tfrac{1}{4}y - 6 = 1$ → coefficient ______, constant ______

   c) $8 - 3n = 20$ → coefficient ______, constant ______

8. In $2x - 7 = 1$, the two operations applied to the variable are ______________ then ______________.

   The two operations that undo them are ______________ then ______________.

   Confirm that $x = 4$ is the solution: _______________________________________________

9. Write a two-step equation of the form $ax + b = c$ whose solution is $x = 3$, then show the check.

   Equation: _______________ Check: _______________________________________________

PAGE 5 — Apply, reason, repair

Think It Through

10. Application. A taxi ride costs a $3.00 pickup fee plus $2.00 per mile, so the total for $m$ miles is $2m + 3 = 13$. Is $m = 5$ a solution?

    Work: _______________________________________________

    What it means about the ride: _______________________________________________

11. Reasoning. Why must a check substitute into the original equation, not into a line written partway through?

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12. Error analysis. Jordan tests $x = -2$ in $-3x + 5 = 11$ and writes $-3(-2) = -6$, then $-6 + 5 = -1$, and concludes that $-2$ is not a solution.

    Jordan's mistake: _______________________________________________

    Correct check: _______________________________________________

    Is $-2$ a solution? ______

PAGE 6 — Exit ticket 10.1

Exit Ticket · Lesson 10.1

Name: ________________________ Date: ____________

13. Is $x = 7$ a solution of $2x - 5 = 9$? Show the substitution.

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14. Which value from $-4$, $0$, and $4$ is the solution of $3x + 10 = -2$? ______

15. In $-6x + 1 = 19$: coefficient ______, constant term ______, undo first _______________

16. What does it mean for a number to be a solution of an equation?

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PAGE 7 — The balance scale

10.2 Modeling with a Balance

FIGURE: fig1-balance-two-step.png (full width)

Write the equation this balance shows. _______________

The rule. Whatever you do to one pan, you must do to the ______________.

FIGURE: fig2-balance-sequence.png (full width)

Label each panel with its equation.

Panel 1: _______________ Panel 2: _______________ Panel 3: _______________

What move takes you from panel 1 to panel 2?

What move takes you from panel 2 to panel 3?

PAGE 8 — Algebra tiles and zero pairs

Tiles and Zero Pairs

FIGURE: fig3-algebra-tiles-two-step.png (full width)

Fill in. The top row models _______________. Removing ______ unit tiles from both mats gives _______________. Splitting each mat into ______ equal groups gives $x =$ ______.

FIGURE: fig4-zero-pairs-tiles.png (full width)

A zero pair is _______________________________________________

Why can you add three $1$ tiles to each mat without changing the equation?

PAGE 9 — Guided practice

Model It

17. A balance holds two blocks labeled $x$ and 5 chips on the left pan, and 13 chips on the right. Write the equation.

    Equation: _______________

18. Solve item 17 by describing the move at each pan, then confirm by rebuilding the scale.

    Move 1: _______________________________________________

    Move 2: _______________________________________________

    $x =$ ______ Confirmation: _______________________________________________

19. Model $4x + 3 = 15$ with algebra tiles.

    Left mat: ______ $x$ tiles and ______ unit tiles. Right mat: ______ unit tiles.

    Move 1: _______________ Move 2: _______________ $x =$ ______

20. What is a zero pair? Why does adding three $1$ tiles to each mat of $2x - 3 = 7$ leave the equation true?

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    Solution: $x =$ ______

Draw your own balance for item 17 in the space below.

BLANK BALANCE FRAME: beam, fulcrum, and two empty pans, full width

PAGE 10 — Independent practice

Models and Moves

21. Write the equation each model shows and solve it.

    a) three blocks labeled $x$ and 2 chips on the left pan, 17 chips on the right

    Equation: _______________ $x =$ ______

    b) two $x$ tiles and four $-1$ tiles on the left mat, six $1$ tiles on the right

    Equation: _______________ $x =$ ______

22. Model $5x + 6 = 21$. Move 1: _______________ Move 2: _______________ $x =$ ______

23. Model $3x - 5 = 7$, naming the zero pairs. Move 1: _______________ Move 2: _______________

    $x =$ ______ Confirmation with the model: _______________________________________________

24. Model $-2x + 9 = 3$. Move 1: _______________ Move 2: _______________ $x =$ ______

25. Model $6 = 2x + 10$. Move 1: _______________ Move 2: _______________ $x =$ ______

PAGE 11 — Repair, apply, reason

Fix It and Use It

26. Error analysis. Devon models $2x + 5 = 13$, removes 5 unit tiles from the left mat only, splits both mats into 2 equal groups, and reports $x = 6.5$.

    What went wrong: _______________________________________________

    Correct solution: $x =$ ______

27. Application. Three identical boxes and a 4-pound weight balance a 22-pound weight.

    Equation with $b$ for one box: _______________

    Scale moves: _______________________________________________

    $b =$ ______ pounds Confirmation: _______________________________________________

28. Reasoning. Why must both mats be split into the same number of equal groups? What would go wrong if you split the left mat into 3 groups and the right into 2?

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PAGE 12 — Exit ticket 10.2

Exit Ticket · Lesson 10.2

Name: ________________________ Date: ____________

29. A balance holds two blocks labeled $x$ and 7 chips on the left pan and 15 chips on the right.

    Equation: _______________ $x =$ ______

30. Model $3x - 2 = 10$ with tiles. Moves: _______________________________________________

    $x =$ ______

31. Use a model to confirm whether $x = 5$ is the solution of $2x + 3 = 13$. ______

32. Why does removing the same number of unit tiles from both mats keep the equation true?

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PAGE 13 — The four properties of equality

10.3 Solving Two-Step Equations

The procedure.

1. Name the ______________ and the ______________ term, signs included.
2. Undo the ______________ or ______________ first, on both sides.
3. Undo the ______________ or ______________ second, on both sides.
4. ______________ by substituting into the original equation.

Complete the table.

PAGE 14 — Guided practice with solution frames

Two Steps, Every Time

33. Solve $2x + 9 = 21$, naming the property at each step.

$$2x + 9 = 21$$ $$2x + 9 - \underline{\quad} = 21 - \underline{\quad} \qquad \text{(\_\_\_\_\_\_\_\_\_\_ property of equality)}$$ $$2x = \underline{\quad}$$ $$\frac{2x}{\underline{\quad}} = \frac{\underline{\quad}}{\underline{\quad}} \qquad \text{(\_\_\_\_\_\_\_\_\_\_ property of equality)}$$ $$x = \underline{\quad}$$

Check: _______________________________________________

34. Solve $5x - 8 = 7$. Step 1: _______________ Step 2: _______________ $x =$ ______ Check: ______

35. Solve $\tfrac{x}{3} + 4 = 9$. Step 1: _______________ Step 2: _______________ $x =$ ______ Check: ______

36. Solve $-6x + 5 = 23$. Step 1: _______________ Step 2: _______________ $x =$ ______ Check: ______

PAGE 15 — Independent practice

Solve and Check

37. Solve and check.

38. Solve and check.

39. Solve and check. The variable is on the right.

    a) $26 = 5x + 6$ → $x =$ ______ Check: ______

    b) $-7 = 3 - 2x$ → $x =$ ______ Check: ______

PAGE 16 — Exact answers, reasoning, repair

When the Answer Is a Fraction

40. Solve $6x + 5 = 12$ and check. State the solution exactly — do not round.

    $6x =$ ______ $x =$ ______ Check: $6\left(\underline{\quad}\right) + 5 = \underline{\quad}$

41. Reasoning. Show every step of $8 - 5x = 43$, naming the property used at each step.

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    $x =$ ______

    Why does undoing in reverse order work? _______________________________________________

42. Write a two-step equation with a negative coefficient whose solution is $x = -2$.

    Equation: _______________ Check: _______________________________________________

43. Application. A gym charges a $25 joining fee plus $18 per month, and the total is $151.

    $18m + 25 = 151$ → $m =$ ______ Sentence: _______________________________________________

44. Error analysis. Priya solves $3x + 12 = 30$ by dividing by 3 first, writing $x + 12 = 10$, then $x = -2$.

    Her mistake: _______________________________________________

    Correct solution: $x =$ ______ Check that exposes her answer: _______________________________________________

PAGE 17 — Exit ticket 10.3

Exit Ticket · Lesson 10.3

Name: ________________________ Date: ____________

45. Solve $5x - 7 = 28$. $x =$ ______ Check: ______

46. Solve $-2x + 11 = 3$. $x =$ ______ Check: ______

47. Solve $\tfrac{x}{6} + 2 = 5$. $x =$ ______ Check: ______

48. Why do you undo addition or subtraction before multiplication or division?

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PAGE 18 — From words to symbols

10.4 Writing an Equation for a Situation

Fixed amount + repeated amount × how many = total, which is $ax + b = c$.

Label the parts of $ax + b = c$.

$x$ is the _______________ $a$ is the _______________

$b$ is the _______________ $c$ is the _______________

Guided practice.

49. A skating rink charges $5 admission plus $2 per hour. Tomas paid $15.

    Let $h =$ _______________ Equation: _______________ $h =$ ______

50. Three more than twice a number is 17. Equation: _______________ $n =$ ______

51. A plumber charges a $60 service fee plus $45 per hour, and the bill was $285.

    Let $h =$ _______________ Equation: _______________ $h =$ ______

52. Write a situation in context for $3x + 8 = 29$.

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    Solution: $x =$ ______

PAGE 19 — Independent practice

Write the Equation

53. Eight less than five times a number is 32. Equation: _______________ $n =$ ______

54. A number divided by 4, then increased by 6, is 10. Equation: _______________ $n =$ ______

55. Application. A food truck charges $2.50 per taco plus a $1.00 packaging fee. An order came to $11.00.

    Let $t =$ _______________ Equation: _______________ $t =$ ______

56. A pool holds 30 inches of water and drains 4 inches per hour. After $h$ hours, 10 inches remain.

    Equation: _______________ $h =$ ______

57. Write a situation in context for $7x + 12 = 61$.

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    Solution: $x =$ ______

PAGE 20 — Write the situation

Turn an Equation into a Story

58. Reasoning. Write a situation in context for $\tfrac{1}{2}x - 3 = 6$.

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    Solution: $x =$ ______

    Why must your story halve the unknown before it subtracts 3?

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59. Write a situation in context for $-3x + 25 = 4$, choosing a setting where a decrease makes sense.

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    Solution: $x =$ ______

60. Error analysis. For "tickets cost $6 each plus a $4 order fee, total $34," Sam writes $6(t + 4) = 34$.

    Why it does not match: _______________________________________________

    Correct equation: _______________ $t =$ ______

PAGE 21 — Exit ticket 10.4

Exit Ticket · Lesson 10.4

Name: ________________________ Date: ____________

61. A fair charges a $7 entry fee plus $3 per ride, and Nia spent $28.

    Equation: _______________ $r =$ ______

62. Six less than three times a number is 9. Equation: _______________ $n =$ ______

63. Write a situation for $4x + 9 = 33$: _______________________________________________

    Solution: $x =$ ______

64. How do you decide which quantity should be the variable?

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PAGE 22 — Finishing a problem in context

10.5 Solving Problems in Context

FIGURE: fig7-bar-model-context.png (full width)

The six-step finish. Check each box as you do it.

☐ Define the variable, with units ☐ Write the equation ☐ Solve, naming your moves ☐ Check in the original equation ☐ Check against the story ☐ State the answer in a sentence with units

Guided practice.

65. A rectangle has a length of 12 cm and a perimeter of 40 cm.

    Let $w =$ _______________ Equation: _______________ $w =$ ______ cm

66. Tia has $50 and earns $12 per lawn. She wants $146.

    Equation: _______________ $n =$ ______ lawns

67. A tank holds 60 liters and loses 7 liters per hour. When do 18 liters remain?

    Equation: _______________ $h =$ ______ hours

68. Five identical sandwiches plus one $2.50 drink came to $32.50.

    Equation: _______________ $s =$ $______

PAGE 23 — Independent practice

Real Situations

69. Application. A studio charges a $40 membership fee plus $8.50 per class. Kara paid $91.

    Equation: _______________ $c =$ ______ Sentence: _______________________________________________

70. A taxi charges $3.25 plus $1.75 per mile. A ride cost $17.25.

    Equation: _______________ $m =$ ______ miles

71. An isosceles triangle has two equal sides of length $s$ and a base of 7 in. Its perimeter is 31 in.

    Equation: _______________ $s =$ ______ in.

72. A diver at an elevation of $-6$ meters rises 1.5 meters per second. When is the diver 3 meters above the surface?

    Equation: _______________ $t =$ ______ seconds

73. A jar holds 3 quarters and some dimes, worth $1.85 in all.

    Equation: _______________ $n =$ ______ dimes

74. A 10-foot board has 1 foot trimmed off; the rest is cut into 4 equal pieces. State the answer exactly.

    Equation: _______________ $p =$ ______ feet

PAGE 24 — Reason and repair

Does the Answer Make Sense?

75. Reasoning. Ana models a savings problem with $6x + 45 = 105$ and finds $x = 10$.

    Substitution check: _______________________________________________

    What else must she do before the answer counts as complete?

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76. Error analysis. For "a $12 service fee plus $5 per hour, total $47," Leo writes $12h + 5 = 47$ and gets $h = 3.5$.

    What he mixed up: _______________________________________________

    Correct equation: _______________ $h =$ ______

    Check against the story: _______________________________________________

PAGE 25 — Exit ticket 10.5

Exit Ticket · Lesson 10.5

Name: ________________________ Date: ____________

77. A rectangle has a length of 15 m and a perimeter of 54 m. Width = ______ m

78. A $20 gift card buys songs at $1.25 each, and $8.75 remains.

    Equation: _______________ $n =$ ______ songs

79. Ravi has $35 and adds $9 each week until he has $116.

    Equation: _______________ $w =$ ______ weeks

80. Why state your answer with units and check it against the story?

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PAGE 26 — Chapter 10 review, part 1

Chapter 10 Review

Part A · Modeling (7.PFA.3a)

81. A balance holds three blocks labeled $x$ and 4 chips on the left pan and 19 chips on the right.

    Equation: _______________ Moves: _______________________________________________ $x =$ ______

82. Model and solve $2x + 6 = 14$ with tiles: _______________________________________________ $x =$ ______

83. Model and solve $3x - 4 = 8$ with tiles, naming the zero pairs: _______________________________ $x =$ ______

84. Model and solve $-2x + 7 = 1$ with tiles: _______________________________________________ $x =$ ______

Part B · Solving with properties (7.PFA.3b)

85. Solve and check.

86. a) $\tfrac{2}{3}x - 5 = 1$ → $x =$ ______ b) $1.5x + 2.5 = 11.5$ → $x =$ ______

87. $30 = 8 - 2x$ → $x =$ ______

88. $10x + 3 = 8$ → $x =$ ______ (exact)

89. Show every step of $-3x + 14 = 2$, naming each property: ______________________________________

PAGE 27 — Chapter 10 review, part 2

Chapter 10 Review (continued)

Part C · Confirming solutions (7.PFA.3c)

90. Is $x = 6$ the solution of $4x - 5 = 19$? ______ Work: _______________

91. Is $x = 3$ the solution of $5x - 4 = 6$? ______ If not, the solution is ______

92. Marcus checks $x = -3$ in $-2x + 5 = 11$ and writes $-2(-3) = -6$.

    His error: _______________________________________________

    Correct check: _______________ Is $-3$ a solution? ______

Part D · Writing equations (7.PFA.3d)

93. Tickets cost $9 each plus a $5 order fee; the total was $68.

    Equation: _______________ $t =$ ______

94. Eleven less than six times a number is 31. Equation: _______________ $n =$ ______

95. A 24-inch candle burns 2 inches per hour; after $h$ hours, 9 inches remain. Exact answer.

    Equation: _______________ $h =$ ______

PAGE 28 — Chapter 10 review, part 3

Chapter 10 Review (continued)

Part E · Writing situations (7.PFA.3e)

96. Write a situation for $5x + 12 = 47$: _______________________________________________

    Solution: $x =$ ______

97. Write a situation for $-4x + 30 = 6$: _______________________________________________

    Solution: $x =$ ______

Part F · Problems in context (7.PFA.3f)

98. A rectangle has a length of 11 ft and a perimeter of 38 ft. Width = ______ ft

99. Kim has $18 and saves $6.50 each week until she has $63.50.

    Equation: _______________ $w =$ ______ weeks

100. A phone plan costs $12 plus $0.25 per gigabyte; the bill was $19.50.

     Equation: _______________ $g =$ ______ gigabytes

PAGE 29 — Chapter 10 review, part 4

Chapter 10 Review · Mixed Application and Reasoning

Part G

101. Solve $3x + 5 = 20$ two ways.

     Route 1 — subtract 5, then divide by 3: _______________________________________________

     Route 2 — divide the entire side by 3, then subtract: _______________________________________

     Why do they agree? _______________________________________________

102. A student says the solution of $-2x + 9 = 1$ is $x = -4$.

     Substitution check: _______________ Correct solution: $x =$ ______

103. Write one two-step equation with a negative coefficient and one with a fractional coefficient, each with solution $x = 6$.

     Negative coefficient: _______________ Check: _______________

     Fractional coefficient: _______________ Check: _______________

104. Plan A costs $30 plus $4 per class. Plan B costs $10 plus $8 per class.

     Plan A costs $70 after ______ classes. Equation: _______________

     Plan B costs $70 after ______ classes. Equation: _______________

     What does the Plan B answer mean when classes cannot be split?

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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
• Figure widths: all seven chapter figures are landscape; place at full width. fig2-balance-sequence.png is the widest — do not shrink it below full column width or the chip labels stop being legible.
• Blank balance frame: page 9 needs an unlabeled balance (level beam, fulcrum, two empty pans) for student drawing; add a blank_balance() variant to figures/make_grade7_ch10.py if a rendered version is wanted rather than a drawn box.
• Solution-step frames: pages 14 and 15 need generous vertical space between equation lines so students can write a full step on each line
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
• Money: every dollar amount is typeset as plain text, not math, so the dollar sign renders as a dollar sign