MathBored

Virginia SOL Mathematics Textbook

Grade 8 Workbook — Chapter 5: Multistep Linear Equations

SOL 8.PFA.4 · Companion to Textbook Chapter 5

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 124.


PAGE 1 — Chapter opener

Chapter 5 · Multistep Linear Equations

Standard 8.PFA.4

In this chapter you will:

Words to know: multistep linear equation · like terms · distributive property · expand · properties of equality · additive inverse property · multiplicative identity property · reciprocal · algebra tiles · zero pair · colored chips · balance scale · bar model · substitution · confirm · interpret · break-even point

Reminder from Grade 7: a two-step equation like 3x+7=223x + 7 = 22 needed two undo moves. Here a side may need simplifying first, and the variable may appear on both sides — but never more than four steps in all.


PAGE 2 — Simplify each side first

5.1 Combining Like Terms Before You Solve

Step 0 of every multistep equation: make each side as short as it can be.

5x+2x3=18x3=185x + 2x - 3 = 18 \quad \rightarrow \quad \underline{\hspace{2cm}}\,x - 3 = 18

Complete the table.

Equation Rewritten as a sum Simplified side
5x+2x3=185x + 2x - 3 = 18
4x+9x=244x + 9 - x = 24
30=8n3n+530 = 8n - 3n + 5
104k+k=110 - 4k + k = 1

Solve. Show the simplified line, then each move, then the check.

  1. 3x+4x5=163x + 4x - 5 = 16

    Simplified: _______________ Solution: x=x = ______

    Check in the original: _______________________________________________

  2. 6a2a+7=276a - 2a + 7 = 27

    Simplified: _______________ Solution: a=a = ______ Check: ______________

  3. 23=9m+34m23 = 9m + 3 - 4m

    Simplified: _______________ Solution: m=m = ______ Check: ______________

  4. 1.2x+0.8x+3=111.2x + 0.8x + 3 = 11

    Simplified: _______________ Solution: x=x = ______ Check: ______________


PAGE 3 — Practice · Simplify, then solve

Practice · Lesson 5.1

  1. Solve and confirm.
a) 2x+5x+4=252x + 5x + 4 = 25 b) 8y3y6=148y - 3y - 6 = 14
c) 104k+k=110 - 4k + k = 1 d) 7=2n+5n217 = 2n + 5n - 21
  1. Solve and confirm.
a) 13x+23x+5=12\tfrac{1}{3}x + \tfrac{2}{3}x + 5 = 12 b) 0.25p+0.75p2.5=4.50.25p + 0.75p - 2.5 = 4.5
  1. 3x+8x11=19-3x + 8x - 11 = 19

    Simplified: _______________ x=x = ______ Check: _______________

  2. 14=62w+5w14 = 6 - 2w + 5w State the solution exactly.

    Simplified: _______________ w=w = ______ Check: _______________

  3. Write an equation that requires combining like terms and whose solution is x=4x = 4.

    Equation: _______________ Check: _______________


PAGE 4 — Application, reasoning, error hunt

Show What You Know · Lesson 5.1

  1. Apply it. A triangle has sides of xx, 2x2x, and 3x+43x + 4 inches, and its perimeter is 34 inches.

    Equation: _______________ Simplified: _______________ x=x = ______

    Three side lengths: ______ in, ______ in, ______ in

  2. Explain. Why does combining like terms on one side require nothing to be done to the other side, even though every property of equality does?



  3. Find the error. Asked to solve 4x+3x2=194x + 3x - 2 = 19, a student writes 5x=195x = 19 on the next line.

    What went wrong? _______________________________________________

    Correct solution: x=x = ______ Check: _______________


PAGE 5 — Exit ticket 5.1

Exit Ticket · Lesson 5.1

Name: ________________________ Date: ____________

  1. 5x+3x7=175x + 3x - 7 = 17 x=x = ______ Check: _______________

  2. 9=4tt+39 = 4t - t + 3 t=t = ______ Check: _______________

  3. 0.5m+2.5m1=110.5m + 2.5m - 1 = 11 m=m = ______ Check: _______________

  4. Why can combining like terms never change the solution of an equation?



PAGE 6 — Blocks in both pans

5.2 Modeling with the Variable on Both Sides

FIGURE: fig1-balance-both-sides.png (full width)

Fill in the blanks.

The left pan holds ______ blocks labeled xx and ______ chips. The right pan holds ______ block labeled xx and ______ chips.

The beam is level, so the equation is _______________.

The one rule: whatever you do to one pan, _______________________________________________.

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

Label each panel with the move that produced it.

Panel 1 → Panel 2: _______________________________________________

Panel 2 → Panel 3: _______________________________________________

Panel 3 → Panel 4: _______________________________________________

Why take the blocks off before the chips? _______________________________________________


PAGE 7 — Tiles, like terms, and zero pairs

Tiles on Two Mats

FIGURE: fig3-tiles-combine-like-terms.png (full width)

Sliding the xx tiles together is the hands-on version of ____________________.

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

A +1+1 tile and a 1-1 tile together are worth ______. They form a ____________.

Guided practice.

  1. A balance holds four xx blocks and 3 chips on the left pan, and one xx block and 12 chips on the right pan.

    Equation: _______________

  2. Solve item 17 by describing each move, then confirm by rebuilding the scale.

    Move 1: _______________ Move 2: _______________ Move 3: _______________

    x=x = ______ Rebuilt check: left ______ chips, right ______ chips

  3. Model 2x+6=4x2x + 6 = 4x with tiles.

    TILE DRAWING SPACE: 1.5 in tall, full width — two mats side by side

    Moves: _______________________________________________ x=x = ______

  4. Model 3x4=x+23x - 4 = x + 2, naming the zero pairs you use.

    Zero pairs used: _______________ x=x = ______ Model check: _______________


PAGE 8 — Practice · Modeling

Practice · Lesson 5.2

  1. Write the equation shown by each model and solve it.
Model Equation Solution
a) five xx blocks and 2 chips on the left pan, three xx blocks and 10 chips on the right pan
b) two xx tiles, three more xx tiles, and one 11 tile on the left mat, sixteen 11 tiles on the right mat
  1. Model 6x=2x+126x = 2x + 12. Moves: _______________ x=x = ______

  2. Model 4x3=2x+54x - 3 = 2x + 5, naming the zero pairs.

    Zero pairs: _______________ x=x = ______ Model check: _______________

  3. Model 12x=2x+312 - x = 2x + 3. Moves: _______________ x=x = ______

  4. Model 5x+4=3x+45x + 4 = 3x + 4. Moves: _______________ x=x = ______

    Why is this still a perfectly good solution? _______________________________________________


PAGE 9 — Reasoning and application

Show What You Know · Lesson 5.2

  1. Find the error. Devon models 3x+5=x+113x + 5 = x + 11, removes one xx tile from the left mat only, and reports 2x+5=x+112x + 5 = x + 11.

    What went wrong? _______________________________________________

    Correct solution: x=x = ______

  2. Apply it. Two identical boxes together with a 5-pound weight balance one box of the same kind together with a 9-pound weight.

    Equation: _______________ Scale moves: _______________________________________________

    b=b = ______ pounds Scale check: left ______ lb, right ______ lb

  3. Explain. Why is removing one xx block from each pan legal even though you do not know what xx is worth?



PAGE 10 — Exit ticket 5.2

Exit Ticket · Lesson 5.2

Name: ________________________ Date: ____________

  1. Balance: three xx blocks and 7 chips on the left pan, one xx block and 15 chips on the right.

    Equation: _______________ x=x = ______

  2. Model 4x1=2x+74x - 1 = 2x + 7. Moves: _______________ x=x = ______

  3. Use a model to confirm whether x=2x = 2 is the solution of 5x+1=3x+55x + 1 = 3x + 5.

    Left count: ______ Right count: ______ Is it a solution? ______

  4. Why must the counts on the two mats match after every move?



PAGE 11 — Expanding

5.3 Expanding with the Distributive Property

a(b+c)=ab+aca(b + c) = ab + ac

The outside factor multiplies EVERY term inside, sign and all.

Two routes for 3(x+4)=213(x + 4) = 21.

Route 1 — expand first: 3x+=213x + \underline{\hspace{2cm}} = 21, so x=x = ______

Route 2 — divide first: x+4=x + 4 = \underline{\hspace{2cm}}, so x=x = ______

When does Route 2 NOT work? _______________________________________________

Complete the sign table.

Expression Expanded
4(x3)-4(x - 3)
(x7)-(x - 7)
12(6x+8)\tfrac{1}{2}(6x + 8)
23(9x6)\tfrac{2}{3}(9x - 6)
2(3x5)-2(3x - 5)

Guided practice.

  1. 4(x+3)=284(x + 3) = 28 Route 1: _______________ Route 2: _______________ x=x = ______

  2. 3(2x1)=213(2x - 1) = 21 Expanded: _______________ x=x = ______ Check: ______

  3. 2(x+5)+3x=302(x + 5) + 3x = 30 Expanded: _______________ Combined: _______________ x=x = ______

  4. 3(x2)=18-3(x - 2) = 18 Expanded: _______________ x=x = ______ Check: ______


PAGE 12 — Practice · Expand, then solve

Practice · Lesson 5.3

  1. Solve and confirm.
a) 5(x2)=255(x - 2) = 25 b) 6(2x+1)=426(2x + 1) = 42
c) 2(x+7)=4-2(x + 7) = 4 d) 23(9x6)=20\tfrac{2}{3}(9x - 6) = 20
  1. Solve and confirm.
a) 4(x+1)+2x=344(x + 1) + 2x = 34 b) 3(x4)x=63(x - 4) - x = 6
  1. 2(x+3)+3(x1)=232(x + 3) + 3(x - 1) = 23

    Expanded: _______________ Combined: _______________ x=x = ______ Check: ______

  2. 0.5(4x+6)=110.5(4x + 6) = 11 x=x = ______ Check: ______


PAGE 13 — Reasoning, application, error hunt

Show What You Know · Lesson 5.3

  1. Explain. Solve 3(x+4)=213(x + 4) = 21 both ways.

    Dividing first: _______________________________________________

    Expanding first: _______________________________________________

    Why must the two routes agree? _______________________________________________

  2. Write an equation that requires the distributive property and has solution x=2x = -2.

    Equation: _______________ Check: _______________

  3. Apply it. Four identical gift bags each hold xx stickers and 3 pencils, and the bags hold 40 items in all.

    Equation: _______________ x=x = ______ What the answer means: _______________

  4. Find the error. Asked to solve 2(x5)=8-2(x - 5) = 8, a student writes 2x10=8-2x - 10 = 8 and reports x=9x = -9.

    What went wrong? _______________________________________________

    Correct solution: x=x = ______ Check that exposes 9-9: _______________


PAGE 14 — Exit ticket 5.3

Exit Ticket · Lesson 5.3

Name: ________________________ Date: ____________

  1. 6(x2)=186(x - 2) = 18 x=x = ______ Check: ______

  2. 2(3x+4)4x=202(3x + 4) - 4x = 20 x=x = ______ Check: ______

  3. 5(x+1)=15-5(x + 1) = 15 x=x = ______ Check: ______

  4. Why must the factor outside the parentheses multiply every term inside?



PAGE 15 — Gathering the variable

5.4 Solving with the Variable on Both Sides

FIGURE: fig5-four-step-order.png (full width)

The four steps, in order.

  1. ____________ any parentheses — distributive property
  2. ____________ like terms within each side
  3. ____________ the variable terms on one side, constants on the other
  4. ____________ by the coefficient

Tip: move the ____________ variable term, so the coefficient you divide by stays positive.

Name the property used at each step of 5x+3=3x+115x + 3 = 3x + 11.

Step Line Property
subtract 3x3x 2x+3=112x + 3 = 11
subtract 33 2x=82x = 8
divide by 22 x=4x = 4

Guided practice.

  1. 6x+1=4x+96x + 1 = 4x + 9 Property at each step: _______________ x=x = ______ Check: ______

  2. 8x5=3x+208x - 5 = 3x + 20 x=x = ______ Check: ______

  3. 3(x2)=x+43(x - 2) = x + 4 Expanded: _______________ x=x = ______ Check: ______

  4. 92x=4x159 - 2x = 4x - 15 x=x = ______ Check: ______


PAGE 16 — Practice · Both sides

Practice · Lesson 5.4

  1. Solve and confirm.
a) 7x+2=5x+147x + 2 = 5x + 14 b) 4x9=x+34x - 9 = x + 3
c) 103x=2x10 - 3x = 2 - x d) 2x+7=5x82x + 7 = 5x - 8
  1. Solve and confirm.
a) 5(x1)=3x+75(x - 1) = 3x + 7 b) 2(2x+3)=3(x+5)2(2x + 3) = 3(x + 5)
  1. 6x+42x=x+196x + 4 - 2x = x + 19 Combined: _______________ x=x = ______ Check: ______

  2. 0.4x+1.2=0.2x+20.4x + 1.2 = 0.2x + 2 x=x = ______ Check: ______

  3. 23x+1=13x+4\tfrac{2}{3}x + 1 = \tfrac{1}{3}x + 4 x=x = ______ Check: ______


PAGE 17 — Reasoning, application, error hunt

Show What You Know · Lesson 5.4

  1. Explain. Solve 4x+6=6x24x + 6 = 6x - 2 twice.

    Subtracting 4x4x: _______________________________________________

    Subtracting 6x6x: _______________________________________________

    Why is moving the smaller variable term usually easier? _______________

  2. Apply it. Studio A charges a $30 registration fee plus $4 per class. Studio B charges $6 per class and no fee.

    Let c=c = _______________ Equation: _______________ c=c = ______

  3. Find the error. Solving 5x+2=2x+145x + 2 = 2x + 14, a student subtracts 2x2x from the left side only and writes 3x+2=143x + 2 = 14.

    What went wrong? _______________________________________________

    Correct solution: x=x = ______ Check: _______________


PAGE 18 — Exit ticket 5.4

Exit Ticket · Lesson 5.4

Name: ________________________ Date: ____________

  1. 9x4=6x+119x - 4 = 6x + 11 x=x = ______ Check: ______

  2. 2(x+5)=4x22(x + 5) = 4x - 2 x=x = ______ Check: ______

  3. 12x=3x+412 - x = 3x + 4 x=x = ______ Check: ______

  4. List the four steps in order and say what each one is for.

    1. ____________________ 2. ____________________

    2. ____________________ 4. ____________________


PAGE 19 — From words to symbols

5.5 Writing Equations and Writing Situations

A comparison puts an expression on each side.

Situation phrase What it becomes
a $30 fee plus $12 per month
$18 per month, no fee
the two totals are equal

Careful with word order. "Five less than three times a number" is ____________, not 53n5 - 3n.

Guided practice.

  1. Rink A: $8 admission plus $3 per hour. Rink B: $5 per hour.

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

  2. Four more than twice a number is the same as the number increased by 10.

    Equation: _______________ n=n = ______ Check: ______

  3. A triangle has sides xx, x+2x + 2, and 2x+12x + 1 inches; perimeter 27 inches.

    Equation: _______________ x=x = ______ Sides: ______, ______, ______

  4. Write a situation in context for 5x+15=8x5x + 15 = 8x.

    Situation: _______________________________________________

    Solution: x=x = ______


PAGE 20 — Practice · Writing equations

Practice · Lesson 5.5

  1. Seven less than five times a number equals three times the number increased by 5.

    Equation: _______________ n=n = ______ Check: ______

  2. Apply it. Plan A: $40 plus $2 per gigabyte. Plan B: $6 per gigabyte.

    Let g=g = _______________ Equation: _______________ g=g = ______

  3. Apply it. Three identical crates with a 12-kg weight balance five identical crates with a 2-kg weight.

    Equation: _______________ c=c = ______ kg Check: ______

  4. A rectangle's length is 4 cm more than its width; perimeter 36 cm.

    Equation: _______________ Width ______ cm Length ______ cm


PAGE 21 — Practice · Writing situations

Practice · Write the Story

A good invented situation passes three tests: the operations match, the numbers land in the right roles, and the answer is a sensible thing to have that many of.

  1. Situation for 6x+20=10x6x + 20 = 10x:



    Solution: x=x = ______ Check: _______________

  2. Situation for 4(x+3)=324(x + 3) = 32 (use identical groups):


    Solution: x=x = ______ Check: _______________

  3. Situation for 2x+4x+5=412x + 4x + 5 = 41:


    Solution: x=x = ______ Check: _______________

  4. Find the error. For "shirts cost $12 each plus a $20 shipping charge at one store, and $16 each with free shipping at another; for how many shirts are the totals equal?" a student writes 12s+20=16s+2012s + 20 = 16s + 20.

    Why does that not match? _______________________________________________

    Correct equation: _______________ s=s = ______


PAGE 22 — Exit ticket 5.5

Exit Ticket · Lesson 5.5

Name: ________________________ Date: ____________

  1. Club A: $18 membership plus $4 per visit. Club B: $6 per visit.

    Equation: _______________ v=v = ______

  2. Three more than four times a number is the same as twice the number increased by 15.

    Equation: _______________ n=n = ______

  3. Write a situation in context for 5(x+2)=405(x + 2) = 40.

    _______________________________________________ Solution: x=x = ______

  4. How do you decide which quantity in a comparison belongs on each side?



PAGE 23 — Finishing a problem in context

5.6 Solving, Confirming, Interpreting

A problem in context is finished when you have all six.

  1. defined the variable, with units
  2. written the equation
  3. solved it
  4. confirmed the solution in the ORIGINAL equation
  5. interpreted the solution in the context
  6. stated the answer in a sentence with units

Confirming vs. interpreting.

Confirming answers: _______________________________________________

Interpreting answers: _______________________________________________

FIGURE: fig7-break-even-graph.png (full width)

The lines cross at c=c = , where each total is $.

For 22 classes or fewer, Plan ______ is cheaper. For 23 or more, Plan ______ is cheaper.

FIGURE: fig6-bar-model-both-sides.png (full width)

Trimming the four equal parts from both bars leaves 60=s60 = \underline{\hspace{2cm}}s, so s=s = ______.


PAGE 24 — Guided practice · Context

Guided Practice · Lesson 5.6

  1. Gym A: $50 joining fee plus $10 per month. Gym B: $20 per month.

    Equation: _______________ m=m = ______ Interpretation sentence: _______________

  2. A rectangle's length is 3 times its width; perimeter 48 cm.

    Equation: _______________ Width ______ cm Length ______ cm Perimeter check: ______

  3. Tank A: 80 L draining 6 L per minute. Tank B: 20 L filling 4 L per minute.

    Equation: _______________ t=t = ______ min Amount in each tank: ______ L

  4. Five concert tickets plus a $4 order fee came to $46.50.

    Equation: _______________ t=$t = \$ ______ Usable as a price? ______ Why? ______


PAGE 25 — Practice · Context

Practice · Lesson 5.6

  1. Apply it. Shop A: $60 setup plus $4 per shirt. Shop B: $10 per shirt.

    Equation: _______________ s=s = ______ shirts Equal total: $______

  2. A rectangle's length is 5 cm more than twice its width; perimeter 64 cm.

    Equation: _______________ Width ______ cm Length ______ cm

  3. Sam has $240 and spends $15 per week; Tara has $120 and saves $15 per week.

    Equation: _______________ w=w = ______ weeks Each has $______

  4. A $25 deposit plus three equal monthly payments totaled $187.

    Equation: _______________ One payment: $______

  5. Six friends each pay the same amount, a $9 coupon is applied, and $51 is paid in all.

    Equation: _______________ One share: $______ Interpretation: _______________

  6. Service A: $3.50 plus $1.25 per mile. Service B: $8.00 plus $0.75 per mile.

    Equation: _______________ m=m = ______ miles Equal fare: $______


PAGE 26 — Reasoning and interpretation

Show What You Know · Lesson 5.6

  1. Explain. A trip needs vans carrying 6 students each, or the same number of vans carrying 4 students each plus 15 students in cars.

    Solve 6v=4v+156v = 4v + 15: v=v = ______

    What the exact answer means: _______________________________________________

    Vans actually needed: ______ Why? _______________________________________________

  2. Find the error. For "Plan A costs $40 plus $5 per class and Plan B costs $9 per class," a student writes 40+5c=9c40 + 5c = 9c, finds c=10c = 10, and concludes "Plan A is always cheaper."

    The equation and the solution are correct. What is wrong with the interpretation?


    Correct conclusion: _______________________________________________


PAGE 27 — Exit ticket 5.6

Exit Ticket · Lesson 5.6

Name: ________________________ Date: ____________

  1. Club A: $24 plus $3 per visit. Club B: $7 per visit.

    Equation: _______________ v=v = ______ Interpretation: _______________

  2. A rectangle's length is 2 m more than its width; perimeter 28 m.

    Width ______ m Length ______ m

  3. Confirm whether x=7x = 7 is the solution of 4x5=2x+94x - 5 = 2x + 9.

    Left side: ______ Right side: ______ Solution? ______

  4. What is the difference between confirming a solution and interpreting one?



PAGE 28 — Chapter 5 review, part 1

Chapter 5 Review

Part A · Modeling with concrete and pictorial representations

  1. Balance: five xx blocks and 2 chips on the left pan, two xx blocks and 11 chips on the right.

    Equation: _______________ Moves: _______________ x=x = ______ Scale check: ______

  2. Model and solve 3x+4=x+103x + 4 = x + 10 with algebra tiles.

    TILE DRAWING SPACE: 1.5 in tall, full width

    Moves: _______________ x=x = ______

  3. Model and solve 4x3=2x+14x - 3 = 2x + 1, naming the zero pairs.

    Zero pairs: _______________ x=x = ______

  4. Model and solve 2(x+3)=102(x + 3) = 10 using two identical groups of tiles.

    TILE DRAWING SPACE: 1.5 in tall, full width

    x=x = ______


PAGE 29 — Chapter 5 review, part 2

Chapter 5 Review (continued)

Part B · Solving with properties of equality

  1. Solve and confirm.
a) 4x+3x8=204x + 3x - 8 = 20 b) 5(x3)=205(x - 3) = 20
c) 6x+5=2x+216x + 5 = 2x + 21 d) 3(x+1)=x+113(x + 1) = x + 11
  1. Solve and confirm.
a) 0.6x+0.4x2=50.6x + 0.4x - 2 = 5 b) 34x+14x+6=15\tfrac{3}{4}x + \tfrac{1}{4}x + 6 = 15
  1. 2(3x4)=4x+62(3x - 4) = 4x + 6 x=x = ______ Check: ______

  2. 94x=2x159 - 4x = 2x - 15 x=x = ______ Check: ______

  3. 5x+2=3x+95x + 2 = 3x + 9 State the solution exactly. x=x = ______ Check: ______

  4. Show every step of 4(x2)+3x=274(x - 2) + 3x = 27 with the property used at each step.

Line Property

PAGE 30 — Chapter 5 review, part 3

Chapter 5 Review (continued)

Part C · Writing an equation from a situation

  1. Shop A: $16 base fee plus $4 per hour. Shop B: $8 per hour.

    Equation: _______________ h=h = ______

  2. Eight less than six times a number equals four times the number increased by 2.

    Equation: _______________ n=n = ______

  3. An isosceles triangle has two equal sides of length ss and a base 3 inches shorter than a leg; perimeter 27 inches.

    Equation: _______________ s=s = ______ in Sides: ______, ______, ______

Part D · Creating a situation from an equation

  1. Situation for 3x+12=5x3x + 12 = 5x: _______________________________________________

    x=x = ______

  2. Situation for 6(x+2)=426(x + 2) = 42: _______________________________________________

    x=x = ______

  3. Situation for 2x+4x+5=412x + 4x + 5 = 41: _______________________________________________

    x=x = ______


PAGE 31 — Chapter 5 review, part 4

Chapter 5 Review (continued)

Part E · Solving problems in context

  1. Jae has $100 and saves $25 per week; Mira has $180 and saves $15 per week.

    Equation: _______________ w=w = ______ weeks Amount: $______

  2. A rectangle's length is 3 cm less than twice its width; perimeter 42 cm.

    Equation: _______________ Width ______ cm Length ______ cm

  3. Four festival tickets plus a $6.50 service fee came to $50.50.

    Equation: _______________ One ticket: $______

Part F · Interpreting algebraic solutions in context

  1. Studio A: $60 plus $3 per class. Studio B: $7 per class.

    c=c = ______ Interpretation sentence: _______________________________________________

    Cheaper for 20 classes: ______ (A: $______ , B: $______ )

  2. Solve 6b=4b+156b = 4b + 15, where bb is a number of buses.

    b=b = ______ Check: ______ Why it cannot be the number ordered: _______________

    Buses actually needed: ______

  3. A student models a timing problem with 5h+20=105h + 20 = 10, where hh is hours since noon, and finds h=2h = -2.

    Check: ______ What the negative solution means: _______________________________________________


PAGE 32 — Chapter 5 review, part 5

Chapter 5 Review (continued)

Part G · Mixed reasoning, error analysis, and enrichment

  1. Solve 6x+9=3x+246x + 9 = 3x + 24 two ways.

    Gathering on the left: _______________ Gathering on the right: _______________

    x=x = ______ Check: _______________

  2. Write one equation needing the distributive property and one needing combining like terms, each with solution x=3x = 3.

    Equation 1: _______________ Check: ______

    Equation 2: _______________ Check: ______

  3. Confirm whether x=3x = 3 is the solution of 5(x1)=2x+45(x - 1) = 2x + 4.

    Left side: ______ Right side: ______ Solution? ______

  4. Find the error. Solving 3(x4)=6-3(x - 4) = 6, a student writes 3x12=6-3x - 12 = 6 and reports x=6x = -6.

    What went wrong? _______________ Correct: x=x = ______ Check exposing 6-6: ______

  5. Enrichment. Try to solve 4x+7=4x24x + 7 = 4x - 2.

    What happens: _______________ Why there is no solution: _______________

  6. Enrichment. Try to solve 3(2x+4)=6x+123(2x + 4) = 6x + 12.

    What happens: _______________ Why every number works: _______________


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