Algebra 1 Workbook — Chapter 2: Multistep and Literal Linear Equations

SOL A.EI.1 (a, b, d, e, f) · Companion to Textbook Chapter 2

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 2 · Multistep and Literal Linear Equations

Standard A.EI.1 (a, b, d, e, f)

In this chapter you will:

Write a linear equation in one variable for a situation described in words
Solve multistep equations and name the property that makes each step legal
Gather the variable when it appears on both sides, and clear fractions and decimals
Decide whether an equation has one solution, no solution, or infinitely many
Rearrange a formula to solve for the variable you actually want
Verify every answer three ways — algebraically, graphically, and with technology

Words to know: linear equation in one variable · like terms · distributive property · commutative property of addition · associative property of addition · additive inverse property · multiplicative inverse property · multiplicative identity property · addition, subtraction, multiplication, division, and substitution properties of equality · reciprocal · clearing fractions · clearing decimals · literal equation · formula · rearrange · solution set · identity · one solution · no solution · infinitely many solutions · verify · intersection · $x$ -intercept · break-even point · interpret

One variable only. Systems of two equations are Chapter 8. Inequalities are Chapter 3. Everything here has exactly one unknown.

The grapher checks your work; it does not do your work. When the algebra and the graph disagree, that is information — go find out which one is wrong.

PAGE 2 — Writing the equation

2.1 From a Situation to an Equation

The five moves.

Read the ______________ situation before writing anything.
Name the unknown in a full sentence, with ______________.
Build each ______________ separately.
Decide what is being set ______________.
Reread the ______________ from your equation.

The two shapes. Complete the table.

Situation	Shape of the equation	Example
A total is known		$1.75m + 2.50 = 16.50$
Two options are compared		$12m + 75 = 27m$

Careful with word order. "Seven less than four times a number" is $\underline{\hspace{2cm}}$ , not $7 - 4n$ .

Guided practice. Define the variable, write the equation, solve, check.

A concert charges $\$9.50$ per ticket plus a $\$4.25$ order fee; an order came to $\$61.25$ .

Let $t =$ _______________ Equation: _______________ $t =$ ______ Check: ______
Five more than three times a number is 26.

Equation: _______________ $n =$ ______ Check: ______
A taxi charges $\$3.00$ plus $\$2.25$ per mile; a ride cost $\$23.25$ .

Let $m =$ _______________ Equation: _______________ $m =$ ______
A rectangle's length is 3 cm less than twice its width; perimeter 54 cm.

Length in terms of $w$ : _______________ Equation: _______________

Width ______ cm Length ______ cm Perimeter check: ______

PAGE 3 — Practice · Writing equations

Practice · Lesson 2.1

Write and solve an equation for each comparison.

a) Maya has $\$120$ and saves $\$18$ per week; Devon has $\$210$ and saves $\$12$ per week.	b) One tree is 42 in tall growing 3 in per year; another is 60 in tall growing 1.5 in per year.

Write and solve an equation.

a) A caterer charges $\$22$ per guest plus a $\$150$ room fee; the bill was $\$1{,}030$ .	b) Three identical crates and a 14 kg toolbox have a mass of 71 kg.

A room is $68^\circ$ F and cools $1.5^\circ$ F per hour, reaching $50^\circ$ F.

Equation: _______________ $h =$ ______ hours Check: ______
Twice the sum of a number and 6 equals 3 less than five times the number.

Equation: _______________ $n =$ ______ Check: ______
A pool holds 4,500 gallons and drains 150 gallons per minute until 1,800 gallons remain.

Equation: _______________ $t =$ ______ minutes Check: ______

PAGE 4 — Application, reasoning, error hunt

Show What You Know · Lesson 2.1

Apply it. Print shop A charges a $\$60$ setup fee plus $\$8$ per shirt; shop B charges $\$12$ per shirt.

Let $s =$ _______________ Equation: _______________

$s =$ ______ shirts Equal total: $______ Check: ______
Explain. A classmate writes $8 - n$ for "eight less than a number."

Why is that wrong? _______________________________________________

Correct expression: _______________

A test that catches this on any translation: _______________________________________________
Find the error. For "Store A charges $\$14$ per book plus $\$6$ shipping; Store B charges $\$17$ per book with free shipping," a student writes $14b + 6 = 17b + 6$ .

Why does that not match the situation? _______________________________________________

Correct equation: _______________ $b =$ ______ Check: ______

PAGE 5 — Exit ticket 2.1

Exit Ticket · Lesson 2.1

Name: ________________________ Date: ____________

A gym charges a $\$40$ joining fee plus $\$25$ per month; a member has paid $\$290$ .

Equation: _______________ $m =$ ______
Four less than six times a number equals twice the number increased by 20.

Equation: _______________ $n =$ ______
A rectangle's length is twice its width; perimeter 96 m.

Equation: _______________ Width ______ m Length ______ m
How do you decide which quantity in a comparison belongs on each side?

PAGE 6 — Naming the property

2.2 Every Move Has a Name

Properties of equality — reasons you may change BOTH sides.

Property	What it says
Addition property of equality
Subtraction property of equality
Multiplication property of equality
Division property of equality
Substitution property of equality

Properties of real numbers — reasons you may rewrite ONE side.

Distributive property: $a(b + c) = \underline{\hspace{2cm}}$

Additive inverse property: $a + (-a) = \underline{\hspace{2cm}}$

Multiplicative identity property: $1 \cdot a = \underline{\hspace{2cm}}$

The order of business.

____________ any parentheses
____________ like terms within each side
undo the ____________ or subtraction
undo the ____________ or division
____________ in the original equation

Name the property at each step of $5x + 3x - 7 = 41$ .

Line	Property
$8x - 7 = 41$
$8x = 48$
$x = 6$

PAGE 7 — Guided practice · Solving with properties

Guided Practice · Lesson 2.2

$7x + 2x - 5 = 40$

Combined: _______________ Property: _______________

$x =$ ______ Check: _______________
$3(x + 4) - 5 = 22$

Expanded: _______________ Property: _______________

$x =$ ______ Check: _______________
$-2(3x - 4) = 26$

Expanded: _______________ What happens to the second sign? _______________

$x =$ ______ Check: _______________
$\frac{3}{4}x - 2 = 10$ Reciprocal used: ______ $x =$ ______ Check: ______

PAGE 8 — Practice · Solve and verify

Practice · Lesson 2.2

Solve and verify.

a) $6x + 5x - 8 = 47$	b) $5(x - 2) = 35$

c) $8 - 3x = 29$	d) $2(4x + 1) - 3x = 27$

Solve and verify.

a) $\frac{1}{2}x + \frac{1}{3}x = 10$	b) $0.25x + 0.75x - 3 = 9$

$-4(x + 2) + 9 = 1$ $x =$ ______ Check: ______

Why is this an ordinary answer and NOT "no solution"? _______________________________________________
$\frac{2}{5}(10x - 15) = 26$ Expanded: _______________ $x =$ ______ Check: ______
$3(2x - 1) + 4(x + 2) = 55$ Expanded: _______________ Combined: _______________ $x =$ ______

PAGE 9 — Application, reasoning, error hunt

Show What You Know · Lesson 2.2

Apply it. A landscaper charges $\$45$ per hour plus $\$120$ for materials; the bill was $\$525$ .

Equation: _______________ $h =$ ______ hours What the answer means: _______________
Explain. Show every step of $5(x - 3) + 2x = 34$ with its property.

Line	Property

Why does combining like terms need no matching move, while subtracting 15 does?

_______________________________________________

Find the error. Solving $-3(x - 6) = 15$ , a student writes $-3x - 18 = 15$ and reports $x = -11$ .

What went wrong? _______________ Correct: $x =$ ______

Check that exposes $-11$ : _______________

PAGE 10 — Exit ticket 2.2

Exit Ticket · Lesson 2.2

Name: ________________________ Date: ____________

$4x + 9x - 6 = 46$ $x =$ ______ Property used to undo the $-6$ : _______________
$6(x + 2) - 5 = 25$ $x =$ ______ Check: ______
$\frac{5}{6}x + 4 = 19$ $x =$ ______ Check: ______
What is the difference between a property of equality and a property of real numbers? Give one example of each from your work above.

PAGE 11 — Both sides, and clearing

2.3 Gathering, Clearing, Solving

The extra step: ____________ the variable terms on one side, using the ____________ or ____________ property of equality.

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

Clearing. Fill in the multiplier.

Equation	Multiply both sides by	Becomes
$\frac{1}{2}x + 7 = \frac{3}{4}x + 1$
$0.15x + 2.4 = 0.05x + 3.6$
$\frac{1}{3}x + 2 = \frac{1}{6}x + 5$

Multiplying both sides is authorized by the ______________________ property of equality, and every term must be multiplied, by the ______________________ property.

Guided practice.

$8x + 5 = 3x + 30$ Property at each step: _______________ $x =$ ______ Check: ______
$2(x + 6) = 5x - 9$ Which variable term did you move, and why? _______________ $x =$ ______
$\frac{1}{3}x + 2 = \frac{1}{6}x + 5$ Cleared: _______________ $x =$ ______ Check in the original: ______
$0.2x + 1.4 = 0.5x - 0.4$ Cleared: _______________ $x =$ ______ Check in the original: ______

PAGE 12 — Practice · Both sides

Practice · Lesson 2.3

Solve and verify.

a) $9x - 7 = 5x + 13$	b) $4(x - 1) = 2x + 10$

c) $12 - 5x = 2 - 3x$	d) $5(x + 3) = 3(x + 7)$

Solve and verify.

a) $\frac{2}{3}x + 1 = \frac{1}{3}x + 6$	b) $0.6x - 1.2 = 0.4x + 2$

$6x + 4 - 2x = x + 22$ Combined: _______________ $x =$ ______ Check: ______
$5 - 3(x - 2) = 2x - 9$ Expanded: _______________ $x =$ ______ Check: ______
$\frac{x + 5}{3} = x - 5$ Cleared: _______________ $x =$ ______ Check: ______

PAGE 13 — Application, reasoning, error hunt

Show What You Know · Lesson 2.3

Apply it. Truck rental A: $\$49$ plus $\$0.60$ per mile. Rental B: $\$29$ plus $\$0.85$ per mile.

Let $m =$ _______________ Equation: _______________

$m =$ ______ miles Equal cost: $______ Check: ______
Explain. Solve $4x + 6 = 6x - 10$ two ways.

Subtracting $4x$ : _______________________________________________

Subtracting $6x$ : _______________________________________________

Why can two legal routes never disagree? _______________________________________________
Find the error. Solving $5x + 2 = 2x + 17$ , a student subtracts $2x$ from the left side only and writes $3x + 2 = 17$ .

Why is that line false even though the final answer comes out right?

Correct solution with properties named: _______________ $x =$ ______

PAGE 14 — Exit ticket 2.3

Exit Ticket · Lesson 2.3

Name: ________________________ Date: ____________

$10x - 3 = 6x + 17$ $x =$ ______ Check: ______
$3(x + 4) = 5x - 2$ $x =$ ______ Check: ______
$\frac{1}{4}x + 3 = \frac{1}{2}x - 1$ Cleared: _______________ $x =$ ______ Check: ______
Why can multiplying both sides by a common denominator never change the solution?

PAGE 15 — Three possible answers

2.4 One Solution, No Solution, Infinitely Many

Complete the table.

Equation	Last line	True or false?	Solution count
$5x - 7 = 2x + 8$
$4x + 9 = 4x - 3$
$3(x + 2) = 3x + 6$

The warning. "The variable disappeared" and "the answer is zero" are ______________ outcomes. In $-4(x + 2) + 9 = 1$ the variable survives to $-4x = 0$ , and $x =$ ______ is a perfectly ordinary single solution.

FIGURE: fig2-one-solution-intersecting.png (half width, left)

FIGURE: fig3-no-solution-parallel.png (half width, right)

One solution: the slopes are ______________, so the lines ______________ once.

No solution: the slopes are ______________ and the $y$ -intercepts are ______________, so the lines are ______________.

FIGURE: fig4-infinitely-many-identical.png (half width, left)

FIGURE: fig5-three-solution-sets-number-line.png (half width, right)

Infinitely many: the slopes are ______________ and the $y$ -intercepts are ______________, so the lines ______________.

On a number line, the three solution sets look like: ______________, ______________, and ______________.

PAGE 16 — Guided practice · Counting solutions

Guided Practice · Lesson 2.4

$2(x + 5) = 2x + 10$

Last line: _______________ Count: _______________

What the last line tells you: _______________________________________________
$7x - 4 = 7x + 1$

Last line: _______________ Count: _______________

The graph of the two sides: _______________________________________________
$8x + 3 = 5x + 18$ Count: _______________ $x =$ ______ Check: ______
$-3(2x - 1) = -6x + 3$ Last line: _______________ Count: _______________

Why does the negative factor not change the reasoning? _______________________________________________

PAGE 17 — Practice · Counting solutions

Practice · Lesson 2.4

Solve and classify.

Equation	Last line	One / none / infinitely many
a) $4(x - 2) = 4x - 8$
b) $9x + 5 = 9x - 5$
c) $6x + 1 = 2x + 21$
d) $5(x + 3) = 5x + 3$

Solve and classify.

Equation	Last line	Count
a) $2x + 3(x + 1) = 5x + 3$
b) $3(2x + 4) = 6x + 10$

$4x + 7 = kx + 7$ has infinitely many solutions when $k =$ ______.

For every other $k$ , the count is ______________ and the solution is $x =$ ______.
$3x + c = 3x + 8$ has infinitely many solutions when $c =$ ______.

Why does every other $c$ give no solution at all — not even one? _______________________________________________
$10 - 2(x + 3) = -2x + 4$ Last line: _______________ Count: _______________

PAGE 18 — Application, reasoning, error hunt

Show What You Know · Lesson 2.4

Apply it. Shop A: $\$5$ per shirt plus $\$40$ setup. Shop B: $\$5$ per shirt plus $\$60$ setup.

Equation: _______________ Last line: _______________ Count: _______________

What this says about the two shops: _______________________________________________
Explain. For $ax + b = cx + d$ , decide the count by comparing $a$ with $c$ and $b$ with $d$ .

Comparison	Count	Your example
$a \neq c$
$a = c$ , $b \neq d$
$a = c$ , $b = d$

Find the error. Solving $3x + 5 = 3x + 5$ , a student reaches $0 = 0$ and writes "no solution, because there is no $x$ in the answer."

What did the student misread? _______________________________________________

Correct classification and reason: _______________________________________________

PAGE 19 — Exit ticket 2.4

Exit Ticket · Lesson 2.4

Name: ________________________ Date: ____________

$5(x - 1) = 5x - 5$ Last line: ______ Count: _______________
$8x + 2 = 8x - 6$ Last line: ______ Count: _______________
$7x - 3 = 4x + 12$ Count: _______________ $x =$ ______
Describe the graph of each side in all three cases, and say which feature decides each one.

One solution: _______________________________________________

No solution: _______________________________________________

Infinitely many: _______________________________________________

PAGE 20 — Solving for a letter

2.5 Literal Equations and Formulas

A literal equation has more than one ______________ in it. A formula is a literal equation people ______________.

FIGURE: fig8-literal-parallel-steps.png (full width)

The two columns use the same properties. Fill them in.

Step	Property
$P - 2l = 2w$
$w = \frac{P - 2l}{2}$

Three habits.

Treat every other letter as a ______________.
Undo in ______________ order.
Divide the ______________ side, not one term.

Check a rearrangement. With $P = 84$ and $l = 25$ : the original gives $w =$ ______, and the rearrangement gives $w =$ ______. Agreement is ______________ but not a proof.

Guided practice.

$d = rt$ for $t$ : $t =$ _______________ Property: _______________ Check with $d = 120$ , $r = 40$ : ______
$A = \frac{1}{2}bh$ for $b$ : $b =$ _______________ Properties: _______________
$P = 2l + 2w$ for $l$ : $l =$ _______________ Check with $P = 30$ , $w = 4$ : ______
$y = mx + b$ for $x$ : $x =$ _______________ Restriction: ______________

PAGE 21 — Practice · Rearranging

Practice · Lesson 2.5

Solve for the specified variable.

a) $C = 2\pi r$ for $r$	b) $V = lwh$ for $h$

c) $I = Prt$ for $t$	d) $ax + by = c$ for $y$

Solve for the specified variable.

a) $A = P + Prt$ for $t$	b) $S = 2\pi r^2 + 2\pi rh$ for $h$

$F = \frac{9}{5}C + 32$ for $C$ : $C =$ _______________

Convert $F = 77$ : $C =$ ______ Check: ______
$3x + 4y = 12$ for $y$ : $y =$ _______________ In $y = mx + b$ form: _______________

PAGE 22 — Application, reasoning, error hunt

Show What You Know · Lesson 2.5

Apply it. $A = \frac{1}{2}h(b_1 + b_2)$ for $h$ : $h =$ _______________

Area 48 sq in, bases 6 in and 10 in: $h =$ ______ in Check: ______
Apply it. Use $w = \frac{P - 2l}{2}$ with $P = 84$ cm and $l = 25$ cm.

$w =$ ______ cm Verify in $P = 2l + 2w$ : ______
Explain. Why does solving $P = 2l + 2w$ for $w$ use the same properties as solving $17 = 12 + 2w$ for $w$ ?

What is harder about the literal version, and why is it not the mathematics?
Find the error. Asked to solve $P = 2l + 2w$ for $w$ , a student writes $w = P - 2l$ .

What went wrong? _______________ Correct: $w =$ _______________

Show the failure with $P = 84$ , $l = 25$ : student's $w =$ ______, which gives $P =$ ______

PAGE 23 — Exit ticket 2.5

Exit Ticket · Lesson 2.5

Name: ________________________ Date: ____________

$V = Bh$ for $B$ : $B =$ _______________
$y - y_1 = m(x - x_1)$ for $m$ : $m =$ _______________ Restriction: ______________
$2x - 5y = 20$ for $y$ : $y =$ _______________
How do you check a rearranged formula, and why is agreement on one set of numbers not a proof?

PAGE 24 — Three ways to verify

2.6 Verifying Algebraically, Graphically, and with Technology

What each check catches.

Method	What you do	What it catches
Algebraically
Graphically
With technology

FIGURE: fig1-verify-by-graphing.png (full width)

The lines meet at ( ______ , ______ ).

The solution is the ______-coordinate: $x =$ ______.

The other coordinate, ______, is _______________________________________________.

FIGURE: fig6-difference-graph-zero.png (full width)

For $3x - 5 = x + 1$ , the difference function is $y = \underline{\hspace{2cm}}$ , and the solution is its ______-intercept, $x =$ ______.

When the algebra and the graph disagree, one of three things is wrong:

_______________, _______________, or _______________.

PAGE 25 — Interpreting in context

Interpreting the Answer

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

Shop A charges $\$60$ plus $\$8$ per shirt: total $=$ _______________

Shop B charges $\$12$ per shirt: total $=$ _______________

The lines cross at ( ______ , ______ ), so the bills are equal at ______ shirts, $______ each.

Below 15 shirts, shop ______ is cheaper. Above 15, shop ______ is cheaper.

Three interpretation questions.

What does the number ______________ or ______________?
Is the ______________ sensible?
Can the quantity ______________ this value?

Guided practice.

Verify that $x = 4$ solves $5x - 3 = 3x + 5$ .

Left side: ______ Right side: ______ Solution? ______ Intersection: ( ______ , ______ )
$2x + 7 = 4x - 1$ $x =$ ______ Check: ______ Graphs meet at ( ______ , ______ )
A grapher shows $y = 3x - 1$ and $y = x + 5$ meeting at $(3, 8)$ .

Equation being solved: _______________ Solution: $x =$ ______ What 8 represents: _______________
Taxi: $\$3.00$ plus $\$2.25$ per mile. Rideshare: $\$5.00$ plus $\$1.75$ per mile.

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

PAGE 26 — Practice · Verifying

Practice · Lesson 2.6

Decide whether the proposed value is a solution.

Question	Left side	Right side	Solution?
a) Is $x = 6$ a solution of $4x - 5 = 2x + 7$ ?
b) Is $x = 2$ a solution of $5x + 1 = 3x + 7$ ?

If b) is not a solution, the solution is  $x =$  ______.

$6x - 4 = 2x + 12$ $x =$ ______ Check: ______ Graphs meet at ( ______ , ______ )
$3(x + 1) = x + 9$ $x =$ ______ Difference function: $y =$ _______________ $x$ -intercept: ______
A grapher shows $y = 2x + 5$ and $y = 2x - 1$ never meeting.

Equation: _______________ Count: _______________ How the picture tells you: _______________

PAGE 27 — Application, reasoning, error hunt

Show What You Know · Lesson 2.6

Apply it. Gym A: $\$75$ joining fee plus $\$12$ per month. Gym B: $\$27$ per month.

Equation: _______________ $m =$ ______ months Equal total: $______

Two years (24 months): A costs $______ , B costs $______ , cheaper: ______
Apply it. One drone at 120 m descends 8 m/s; another at 40 m climbs 2 m/s.

Equation: _______________ $t =$ ______ seconds Height: ______ m Check: ______
Explain. Your algebra gives $x = 7$ , your grapher crosses at $x = 7$ , but substitution into the original makes the sides differ.

What must have happened? _______________________________________________

The order in which you would check things: 1. _______________ 2. _______________ 3. _______________
Find the error. Solving $4(x - 1) = 2x + 6$ , a student reports $x = 2$ and says the graph confirms it.

Substitution: left ______ , right ______ , so ______

Correct solution with properties: _______________ $x =$ ______ True intersection: ( ______ , ______ )

PAGE 28 — Exit ticket 2.6

Exit Ticket · Lesson 2.6

Name: ________________________ Date: ____________

$7x - 2 = 4x + 13$ $x =$ ______ Left side: ______ Right side: ______
A grapher shows $y = -x + 8$ and $y = 3x$ meeting at $(2, 6)$ .

Equation: _______________ Solution: $x =$ ______
Photo lab A: $\$25$ plus $\$1.50$ per print. Lab B: $\$2.00$ per print.

Equation: _______________ $p =$ ______ prints Equal cost: $______
What does each verification method catch that the other two might miss?

Algebraic: _______________ Graphical: _______________ Technology: _______________

PAGE 29 — Chapter 2 review, part 1

Chapter 2 Review

Part A · Writing an equation for a contextual situation

A caterer charges $\$18.75$ per plate plus a $\$95$ room fee; the bill was $\$1{,}032.50$ .

Let $p =$ _______________ Equation: _______________ $p =$ ______ plates
Nine less than five times a number is the same as the number increased by 15.

Equation: _______________ $n =$ ______
A rectangle's length is 4 m less than three times its width; perimeter 72 m.

Equation: _______________ Width ______ m Length ______ m
Service A: $\$120$ installation plus $\$55$ per month. Service B: $\$75$ per month.

Equation: _______________ $m =$ ______ months Equal total: $______

PAGE 30 — Chapter 2 review, part 2

Chapter 2 Review (continued)

Part B · Solving multistep equations with named properties

Solve and verify.

a) $7x + 3x - 12 = 38$	b) $4(x - 5) + 6 = 18$

c) $-2(3x + 7) = 10$	d) $9x - 4 = 5x + 24$

Solve and verify.

a) $\frac{3}{5}x + 4 = 19$	b) $0.3x + 1.5 = 0.1x + 3.1$

$5(x + 2) - 3(x - 1) = 27$ Expanded: _______________ $x =$ ______ Check: ______
$\frac{2x - 1}{3} = x - 4$ Cleared: _______________ $x =$ ______ Check: ______
$6 - 4(x - 3) = 2x + 6$ Expanded: _______________ $x =$ ______ Check: ______
Show every step of $3(2x - 5) + 4x = 35$ with its property.

Line	Property

PAGE 31 — Chapter 2 review, part 3

Chapter 2 Review (continued)

Part C · Rearranging a formula or literal equation

$A = bh$ for $h$ : $h =$ _______________
$P = 2(l + w)$ for $w$ : $w =$ _______________
$y = mx + b$ for $b$ : $b =$ _______________
$V = \frac{1}{3}Bh$ for $B$ : $B =$ _______________
$5x - 2y = 10$ for $y$ : $y =$ _______________
$A = P(1 + rt)$ for $r$ : $r =$ _______________

With $A = 1120$ , $P = 1000$ , $t = 2$ : $r =$ ______ Verify in the original: ______

Part D · Determining the number of solutions

Solve and classify.

Equation	Last line	Count
a) $3(x + 5) = 3x + 15$
b) $6x - 1 = 6x + 4$
c) $7x + 2 = 3x + 26$

$4(2x - 3) = 8x - 12$ Last line: ______ Count: _______________
$2(x + 8) = 2x + 8$ Last line: ______ Count: _______________

PAGE 32 — Chapter 2 review, part 4

Chapter 2 Review (continued)

$6x + 5 = kx + 5$ has infinitely many solutions when $k =$ ______.

For every other $k$ : count ______________, solution $x =$ ______
Describe the graphs for all three cases.

Count	What the two lines do	Deciding feature
One solution
No solution
Infinitely many

Explain. A student solving $5x + 3 = 5x + 3$ reaches $3 = 3$ and writes "no solution."

The misreading: _______________________________________________

Correct classification: _______________ The graph that settles it: _______________

Part E · Verifying and interpreting

Is $x = 7$ the solution of $6x - 8 = 4x + 6$ ?

Left side: ______ Right side: ______ Solution? ______
$5x + 2 = 2x + 17$ $x =$ ______ Check: ______ Graphs meet at ( ______ , ______ )
$4(x - 2) = 2x$ $x =$ ______ Difference function: $y =$ _______________ $x$ -intercept: ______

PAGE 33 — Chapter 2 review, part 5

Chapter 2 Review (continued)

A grapher shows $y = 3x + 4$ and $y = 3x - 2$ never meeting.

Equation: _______________ Count: _______________

The algebraic last line that says the same thing: _______________
Apply it. Rideshare A: $\$4.00$ plus $\$1.20$ per mile. Rideshare B: $\$2.50$ plus $\$1.50$ per mile.

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

Check: _______________

12-mile trip: A costs $______ , B costs $______ , cheaper: ______ Why: _______________
Explain. Your algebra gives $x = 6$ ; the grapher's intersection looks like about $x = 6.5$ .

Why not simply pick one? _______________________________________________

Three things that could be wrong: 1. _______________ 2. _______________ 3. _______________

How you would find out which: _______________________________________________

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
Property tables: the "Property" column is the graded column in this chapter, because A.EI.1b asks for the property by name. Make that column at least as wide as the "Line" column so a full phrase such as "subtraction property of equality" fits on one line without wrapping.
Work space: pages 8, 12, 21, 26, and 30 carry the solve-and-verify grids. Give every table cell a minimum height of 1.25 in so a full multistep solution and its check fit inside the cell. Do not shrink type to fit; add a row instead.
Graph paper blocks: on pages 24 and 25, place a 2.5 in square grid (quarter-inch ruling, hairline blue) beside items 82 and 84 so students can sketch the two sides by hand before checking with technology.
Figure widths: fig1-verify-by-graphing.png, fig6-difference-graph-zero.png, fig7-break-even-context.png, and fig8-literal-parallel-steps.png are full width. The four solution-count figures on page 15 run two to a row, half width; keep fig5-three-solution-sets-number-line.png on the right of its row, since it is the widest and shortest of the set.
Color convention: blue is the first function and red the second, matching the chapter figures. If pages are printed in grayscale, add the note "the steeper line is the first function" beneath each two-line graph.
Check lines: every "Check: ______" blank is deliberate. A.EI.1f makes verification its own graded skill, 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.