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:
- Model multistep equations with balance scales and algebra tiles, with the variable on one or both sides
- Simplify a side first, by combining like terms or expanding with the distributive property
- Solve equations of up to four steps using the properties of equality
- Write an equation for a situation, and write a situation for an equation
- Solve problems in context, then confirm and interpret the answer
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 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.
Complete the table.
| Equation | Rewritten as a sum | Simplified side |
|---|---|---|
Solve. Show the simplified line, then each move, then the check.
Simplified: _______________ Solution: ______
Check in the original: _______________________________________________
Simplified: _______________ Solution: ______ Check: ______________
Simplified: _______________ Solution: ______ Check: ______________
Simplified: _______________ Solution: ______ Check: ______________
PAGE 3 — Practice · Simplify, then solve
Practice · Lesson 5.1
- Solve and confirm.
| a) | b) |
|---|---|
| c) | d) |
- Solve and confirm.
| a) | b) |
|---|---|
Simplified: _______________ ______ Check: _______________
State the solution exactly.
Simplified: _______________ ______ Check: _______________
Write an equation that requires combining like terms and whose solution is .
Equation: _______________ Check: _______________
PAGE 4 — Application, reasoning, error hunt
Show What You Know · Lesson 5.1
Apply it. A triangle has sides of , , and inches, and its perimeter is 34 inches.
Equation: _______________ Simplified: _______________ ______
Three side lengths: ______ in, ______ in, ______ in
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?
Find the error. Asked to solve , a student writes on the next line.
What went wrong? _______________________________________________
Correct solution: ______ Check: _______________
PAGE 5 — Exit ticket 5.1
Exit Ticket · Lesson 5.1
Name: ________________________ Date: ____________
______ Check: _______________
______ Check: _______________
______ Check: _______________
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 and ______ chips. The right pan holds ______ block labeled 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 tiles together is the hands-on version of ____________________.
FIGURE: fig4-tiles-both-sides-zero-pairs.png (full width)
A tile and a tile together are worth ______. They form a ____________.
Guided practice.
A balance holds four blocks and 3 chips on the left pan, and one block and 12 chips on the right pan.
Equation: _______________
Solve item 17 by describing each move, then confirm by rebuilding the scale.
Move 1: _______________ Move 2: _______________ Move 3: _______________
______ Rebuilt check: left ______ chips, right ______ chips
Model with tiles.
TILE DRAWING SPACE: 1.5 in tall, full width — two mats side by sideMoves: _______________________________________________ ______
Model , naming the zero pairs you use.
Zero pairs used: _______________ ______ Model check: _______________
PAGE 8 — Practice · Modeling
Practice · Lesson 5.2
- Write the equation shown by each model and solve it.
| Model | Equation | Solution |
|---|---|---|
| a) five blocks and 2 chips on the left pan, three blocks and 10 chips on the right pan | ||
| b) two tiles, three more tiles, and one tile on the left mat, sixteen tiles on the right mat |
Model . Moves: _______________ ______
Model , naming the zero pairs.
Zero pairs: _______________ ______ Model check: _______________
Model . Moves: _______________ ______
Model . Moves: _______________ ______
Why is this still a perfectly good solution? _______________________________________________
PAGE 9 — Reasoning and application
Show What You Know · Lesson 5.2
Find the error. Devon models , removes one tile from the left mat only, and reports .
What went wrong? _______________________________________________
Correct solution: ______
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: _______________________________________________
______ pounds Scale check: left ______ lb, right ______ lb
Explain. Why is removing one block from each pan legal even though you do not know what is worth?
PAGE 10 — Exit ticket 5.2
Exit Ticket · Lesson 5.2
Name: ________________________ Date: ____________
Balance: three blocks and 7 chips on the left pan, one block and 15 chips on the right.
Equation: _______________ ______
Model . Moves: _______________ ______
Use a model to confirm whether is the solution of .
Left count: ______ Right count: ______ Is it a solution? ______
Why must the counts on the two mats match after every move?
PAGE 11 — Expanding
5.3 Expanding with the Distributive Property
The outside factor multiplies EVERY term inside, sign and all.
Two routes for .
Route 1 — expand first: , so ______
Route 2 — divide first: , so ______
When does Route 2 NOT work? _______________________________________________
Complete the sign table.
| Expression | Expanded |
|---|---|
Guided practice.
Route 1: _______________ Route 2: _______________ ______
Expanded: _______________ ______ Check: ______
Expanded: _______________ Combined: _______________ ______
Expanded: _______________ ______ Check: ______
PAGE 12 — Practice · Expand, then solve
Practice · Lesson 5.3
- Solve and confirm.
| a) | b) |
|---|---|
| c) | d) |
- Solve and confirm.
| a) | b) |
|---|---|
Expanded: _______________ Combined: _______________ ______ Check: ______
______ Check: ______
PAGE 13 — Reasoning, application, error hunt
Show What You Know · Lesson 5.3
Explain. Solve both ways.
Dividing first: _______________________________________________
Expanding first: _______________________________________________
Why must the two routes agree? _______________________________________________
Write an equation that requires the distributive property and has solution .
Equation: _______________ Check: _______________
Apply it. Four identical gift bags each hold stickers and 3 pencils, and the bags hold 40 items in all.
Equation: _______________ ______ What the answer means: _______________
Find the error. Asked to solve , a student writes and reports .
What went wrong? _______________________________________________
Correct solution: ______ Check that exposes : _______________
PAGE 14 — Exit ticket 5.3
Exit Ticket · Lesson 5.3
Name: ________________________ Date: ____________
______ Check: ______
______ Check: ______
______ Check: ______
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.
- ____________ any parentheses — distributive property
- ____________ like terms within each side
- ____________ the variable terms on one side, constants on the other
- ____________ by the coefficient
Tip: move the ____________ variable term, so the coefficient you divide by stays positive.
Name the property used at each step of .
| Step | Line | Property |
|---|---|---|
| subtract | ||
| subtract | ||
| divide by |
Guided practice.
Property at each step: _______________ ______ Check: ______
______ Check: ______
Expanded: _______________ ______ Check: ______
______ Check: ______
PAGE 16 — Practice · Both sides
Practice · Lesson 5.4
- Solve and confirm.
| a) | b) |
|---|---|
| c) | d) |
- Solve and confirm.
| a) | b) |
|---|---|
Combined: _______________ ______ Check: ______
______ Check: ______
______ Check: ______
PAGE 17 — Reasoning, application, error hunt
Show What You Know · Lesson 5.4
Explain. Solve twice.
Subtracting : _______________________________________________
Subtracting : _______________________________________________
Why is moving the smaller variable term usually easier? _______________
Apply it. Studio A charges a $30 registration fee plus $4 per class. Studio B charges $6 per class and no fee.
Let _______________ Equation: _______________ ______
Find the error. Solving , a student subtracts from the left side only and writes .
What went wrong? _______________________________________________
Correct solution: ______ Check: _______________
PAGE 18 — Exit ticket 5.4
Exit Ticket · Lesson 5.4
Name: ________________________ Date: ____________
______ Check: ______
______ Check: ______
______ Check: ______
List the four steps in order and say what each one is for.
____________________ 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 .
Guided practice.
Rink A: $8 admission plus $3 per hour. Rink B: $5 per hour.
Let _______________ Equation: _______________ ______
Four more than twice a number is the same as the number increased by 10.
Equation: _______________ ______ Check: ______
A triangle has sides , , and inches; perimeter 27 inches.
Equation: _______________ ______ Sides: ______, ______, ______
Write a situation in context for .
Situation: _______________________________________________
Solution: ______
PAGE 20 — Practice · Writing equations
Practice · Lesson 5.5
Seven less than five times a number equals three times the number increased by 5.
Equation: _______________ ______ Check: ______
Apply it. Plan A: $40 plus $2 per gigabyte. Plan B: $6 per gigabyte.
Let _______________ Equation: _______________ ______
Apply it. Three identical crates with a 12-kg weight balance five identical crates with a 2-kg weight.
Equation: _______________ ______ kg Check: ______
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.
Situation for :
Solution: ______ Check: _______________
Situation for (use identical groups):
Solution: ______ Check: _______________
Situation for :
Solution: ______ Check: _______________
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 .
Why does that not match? _______________________________________________
Correct equation: _______________ ______
PAGE 22 — Exit ticket 5.5
Exit Ticket · Lesson 5.5
Name: ________________________ Date: ____________
Club A: $18 membership plus $4 per visit. Club B: $6 per visit.
Equation: _______________ ______
Three more than four times a number is the same as twice the number increased by 15.
Equation: _______________ ______
Write a situation in context for .
_______________________________________________ Solution: ______
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.
- defined the variable, with units
- written the equation
- solved it
- confirmed the solution in the ORIGINAL equation
- interpreted the solution in the context
- 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 , 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 , so ______.
PAGE 24 — Guided practice · Context
Guided Practice · Lesson 5.6
Gym A: $50 joining fee plus $10 per month. Gym B: $20 per month.
Equation: _______________ ______ Interpretation sentence: _______________
A rectangle's length is 3 times its width; perimeter 48 cm.
Equation: _______________ Width ______ cm Length ______ cm Perimeter check: ______
Tank A: 80 L draining 6 L per minute. Tank B: 20 L filling 4 L per minute.
Equation: _______________ ______ min Amount in each tank: ______ L
Five concert tickets plus a $4 order fee came to $46.50.
Equation: _______________ ______ Usable as a price? ______ Why? ______
PAGE 25 — Practice · Context
Practice · Lesson 5.6
Apply it. Shop A: $60 setup plus $4 per shirt. Shop B: $10 per shirt.
Equation: _______________ ______ shirts Equal total: $______
A rectangle's length is 5 cm more than twice its width; perimeter 64 cm.
Equation: _______________ Width ______ cm Length ______ cm
Sam has $240 and spends $15 per week; Tara has $120 and saves $15 per week.
Equation: _______________ ______ weeks Each has $______
A $25 deposit plus three equal monthly payments totaled $187.
Equation: _______________ One payment: $______
Six friends each pay the same amount, a $9 coupon is applied, and $51 is paid in all.
Equation: _______________ One share: $______ Interpretation: _______________
Service A: $3.50 plus $1.25 per mile. Service B: $8.00 plus $0.75 per mile.
Equation: _______________ ______ miles Equal fare: $______
PAGE 26 — Reasoning and interpretation
Show What You Know · Lesson 5.6
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 : ______
What the exact answer means: _______________________________________________
Vans actually needed: ______ Why? _______________________________________________
Find the error. For "Plan A costs $40 plus $5 per class and Plan B costs $9 per class," a student writes , finds , 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: ____________
Club A: $24 plus $3 per visit. Club B: $7 per visit.
Equation: _______________ ______ Interpretation: _______________
A rectangle's length is 2 m more than its width; perimeter 28 m.
Width ______ m Length ______ m
Confirm whether is the solution of .
Left side: ______ Right side: ______ Solution? ______
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
Balance: five blocks and 2 chips on the left pan, two blocks and 11 chips on the right.
Equation: _______________ Moves: _______________ ______ Scale check: ______
Model and solve with algebra tiles.
TILE DRAWING SPACE: 1.5 in tall, full widthMoves: _______________ ______
Model and solve , naming the zero pairs.
Zero pairs: _______________ ______
Model and solve using two identical groups of tiles.
TILE DRAWING SPACE: 1.5 in tall, full width______
PAGE 29 — Chapter 5 review, part 2
Chapter 5 Review (continued)
Part B · Solving with properties of equality
- Solve and confirm.
| a) | b) |
|---|---|
| c) | d) |
- Solve and confirm.
| a) | b) |
|---|---|
______ Check: ______
______ Check: ______
State the solution exactly. ______ Check: ______
Show every step of 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
Shop A: $16 base fee plus $4 per hour. Shop B: $8 per hour.
Equation: _______________ ______
Eight less than six times a number equals four times the number increased by 2.
Equation: _______________ ______
An isosceles triangle has two equal sides of length and a base 3 inches shorter than a leg; perimeter 27 inches.
Equation: _______________ ______ in Sides: ______, ______, ______
Part D · Creating a situation from an equation
Situation for : _______________________________________________
______
Situation for : _______________________________________________
______
Situation for : _______________________________________________
______
PAGE 31 — Chapter 5 review, part 4
Chapter 5 Review (continued)
Part E · Solving problems in context
Jae has $100 and saves $25 per week; Mira has $180 and saves $15 per week.
Equation: _______________ ______ weeks Amount: $______
A rectangle's length is 3 cm less than twice its width; perimeter 42 cm.
Equation: _______________ Width ______ cm Length ______ cm
Four festival tickets plus a $6.50 service fee came to $50.50.
Equation: _______________ One ticket: $______
Part F · Interpreting algebraic solutions in context
Studio A: $60 plus $3 per class. Studio B: $7 per class.
______ Interpretation sentence: _______________________________________________
Cheaper for 20 classes: ______ (A: $______ , B: $______ )
Solve , where is a number of buses.
______ Check: ______ Why it cannot be the number ordered: _______________
Buses actually needed: ______
A student models a timing problem with , where is hours since noon, and finds .
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
Solve two ways.
Gathering on the left: _______________ Gathering on the right: _______________
______ Check: _______________
Write one equation needing the distributive property and one needing combining like terms, each with solution .
Equation 1: _______________ Check: ______
Equation 2: _______________ Check: ______
Confirm whether is the solution of .
Left side: ______ Right side: ______ Solution? ______
Find the error. Solving , a student writes and reports .
What went wrong? _______________ Correct: ______ Check exposing : ______
Enrichment. Try to solve .
What happens: _______________ Why there is no solution: _______________
Enrichment. Try to solve .
What happens: _______________ Why every number works: _______________
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
- Tile drawing spaces: plain white boxes with a hairline border, split down the middle by a light vertical rule so students can see the two mats. Keep the height called out on each page so a full model fits on one line.
- Color convention: blue = positive, red = negative, matching the chapter figures. If pages are printed in grayscale, add the instruction "shade negative pieces" wherever tiles are drawn.
- Figure widths: every figure in this chapter is full width.
fig2-balance-both-sides-sequence.pngis the widest andfig4-tiles-both-sides-zero-pairs.pngthe tallest; give each its own block with no text wrapping beside it. - Check lines: every "Check: ______" blank is deliberate. Confirming a solution is its own graded skill in this standard (8.PFA.4g), so keep the blank even where space is tight.
- Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.