VertexMath · MTH1W · Free worksheet

Expressions and Equations Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Core Skills (K)

Expressions · Like Terms · Two-Step · Multi-Step · Formulas

K1 Words to Algebra

a)
Write an expression for 7 less than three times a number $n$ .
b)
Write an expression for twice the sum of a number $k$ and $4$ .
c)
Evaluate $-x^2+5x$ for $x=-3$ .

Stuck? Peek at a hint

In (a), ask what the 7 is subtracted from. In (c), write $(-3)$ in brackets everywhere $x$ appears.

K2 Simplify

a)
Simplify $4x^2+3x-x^2+5-7x$ . Then state the number of terms and the coefficient of $x$ in your answer.
b)
Simplify $-3(2a-5)+4a$ .
c)
Simplify $5(y-2)-2(3y+4)$ .

Stuck? Peek at a hint

Expand each bracket first, keeping the sign in front of it with the number you multiply by.

K3 Two-Step Equations

Solve each equation.

a)
$\dfrac{n}{4}-3=2$
b)
$5x+8=-12$
c)
$7-3x=22$

Stuck? Peek at a hint

Undo the addition or subtraction first, then the multiplication or division.

K4 Multi-Step Equations

Solve each equation.

a)
$-3(x-4)=2x-3$
b)
$4x+7=9x-18$
c)
$\dfrac{x-1}{3}+2=\dfrac{x}{2}$

Stuck? Peek at a hint

Expand brackets first. To clear fractions, multiply every term on both sides by the LCD.

K5 Rearranging Formulas

a)
Solve $y=3x-6$ for $x$ .
b)
Solve $A=\dfrac{bh}{2}$ for $h$ . Then find $h$ when $A=30$ cm² and $b=12$ cm.
c)
The angles inside a polygon with $n$ sides add to $S=180(n-2)$ degrees. Solve for $n$ .

Stuck? Peek at a hint

Treat the other letters as numbers and undo the operations on your variable in reverse order.

Real World (A)

Equations in Context · Perimeter · Comparing Plans

A1 Canoe Rental

An outfitter in Algonquin Park rents canoes for $15 plus $12 for each hour.

a)
Write an equation for the total cost $C$ , in dollars, of renting a canoe for $h$ hours. Then find the cost of a 3-hour rental.
b)
A group paid $75. For how many hours did they rent the canoe?
c)
The outfitter charges any part of an hour as a full hour. You have $70. Solve $12h+15=70$ , then state the longest rental you can afford.

Stuck? Peek at a hint

The $15 is paid once; the $12 is paid for every hour.

A2 Community Garden

A rectangular community garden in Hamilton is enclosed by 62 m of fencing. Its length is 4 m more than twice its width $w$ .

a)
Write an equation for the perimeter and solve it. State the width and the length.
b)
Compost is spread at 2 kg per square metre. How much compost does the garden need?
c)
The council asks for a design with the same 62 m of fencing where the length is 35 m more than the width. Solve for the width. What does your answer tell the council?

Stuck? Peek at a hint

Perimeter is the distance all the way around: count all four sides.

A3 Phone Plans

The table shows the monthly cost, in dollars, of two phone plans for $g$ gigabytes (GB) of data. Each plan charges a fixed fee plus a rate per GB.

$\text{Data (GB)}$	$0$	$2$	$4$	$6$
$\text{Plan A cost}$	$25$	$29$	$33$	$37$
$\text{Plan B cost}$	$40$	$41$	$42$	$43$

a)
Write an equation for the cost of each plan, $A$ and $B$ , in terms of $g$ .
b)
Solve $A=B$ . For how much data do the plans cost the same, and what is that cost?
c)
Plan B allows at most 8 GB per month. Can a Plan B customer ever pay the same as on Plan A? Which plan is cheaper for every amount of data Plan B allows?

Stuck? Peek at a hint

The cost at 0 GB is the fixed fee. Use the change in cost for each extra 2 GB to find the rate per GB.

Think It Through (T)

Error Analysis · Counterexamples

T1 Spot the Flaw

A student solves $5-2(x-3)=4x-1$ :
"First $5-2=3$ , so $3(x-3)=4x-1$ . Then $3x-9=4x-1$ , so $x=-8$ ."

a)
Substitute $x=-8$ into both sides of the original equation. Is it a solution?
b)
Identify the student's error and explain why the step is not allowed.
c)
Solve the equation correctly and verify your answer.

Stuck? Peek at a hint

BEDMAS: what has to happen to $-2(x-3)$ before anything is subtracted from $5$ ?

T2 One Test Isn't Proof

Three students solve the rectangle formula $P=2(l+w)$ for $w$ .
Ava: $w=\dfrac{P}{2}-l$ · Jun: $w=\dfrac{P-2l}{2}$ · Noor: $w=P-l-14$

a)
Test all three formulas on a rectangle with $l=10$ cm and $w=4$ cm. Which formulas give the correct width?
b)
Find a rectangle that proves Noor's formula is wrong.
c)
Show that Ava's and Jun's formulas are equivalent. Explain why one successful test can't prove a formula is correct.

Stuck? Peek at a hint

For (b), try a much smaller rectangle and compare Noor's result with its real width.

Boss Level

Optional challenge

There and Back

Maya cycles 30 km along the Niagara River Recreation Trail at 20 km/h, then rides the same 30 km back. Average speed for a trip is the total distance divided by the total time.

a)
She rides back at 30 km/h. Find her average speed for the whole trip.
b)
At what speed must she ride back for her average speed over the whole trip to be 30 km/h?
c)
Is there a return speed that makes her average speed 40 km/h? Justify with an equation.

Stuck? Peek at a hint

Find the time each leg takes before you think about speed.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $3n-7$
b) $2(k+4)$
c) $-24$

a) $3x^2-4x+5$ ; $3$ terms; coefficient of $x$ : $-4$
b) $-2a+15$
c) $-y-18$

a) $n=20$
b) $x=-4$
c) $x=-5$

a) $x=3$
b) $x=5$
c) $x=10$

a) $x=\dfrac{y+6}{3}$
b) $h=\dfrac{2A}{b}$ ; $h=5$ cm
c) $n=\dfrac{S}{180}+2$

Real World

a) $C=12h+15$ ; $51
b) $5$ hours
c) $h\approx4.58$ ; $4$ hours ($63), since a 5-hour rental costs $75.

a) $2w+2(2w+4)=62$ ; width $9$ m, length $22$ m
b) $198$ m² of garden, so $396$ kg
c) $w=-2$ : impossible, a width can't be negative. The two long sides alone would need more than $70$ m of fencing, so no such garden exists.

a) $A=2g+25$ ; $B=0.5g+40$
b) $g=10$ GB; both cost $45
c) No: $g=10$ is past the 8 GB limit. Plan A is cheaper for all data from 0 to 8 GB (at 8 GB: $41 vs $44).

Think It Through

a) Left side $27$ , right side $-33$ : not a solution.
b) $5-2=3$ breaks the order of operations: the $-2$ multiplies the bracket, so it must be distributed before subtracting. $5-2(x-3)=5-2x+6$ .
c) $x=2$ ; both sides equal $7$ .

a) $P=28$ ; all three give $w=4$ .
b) e.g. $l=5$ , $w=2$ , $P=14$ : Noor's formula gives $14-5-14=-5$ , not $2$ . (Any rectangle whose perimeter is not $28$ works.)
c) $\dfrac{P-2l}{2}=\dfrac{P}{2}-\dfrac{2l}{2}=\dfrac{P}{2}-l$ . A test checks one case only: Noor's formula agrees with the correct one only when $P=28$ .

Boss Level

Boss

a) $24$ km/h ( $60$ km in $2.5$ h)
b) $60$ km/h (the return must take $0.5$ h)
c) No: $60$ km at $40$ km/h takes $1.5$ h, so $1.5+t=1.5$ gives a return time $t=0$ , which is impossible.

Free cheat sheet

Expressions and Equations Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

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Most testedSolve $3(x-2)=x+8$ and check your answer. (card 6)

1Words to algebra

For a number $n$ :

$3$ more than $n$	$n+3$
$3$ less than $n$	$n-3$
twice the sum of $n$ and $3$	$2(n+3)$
$n$ squared	$n^2$ , not $2n$

Trap"10 less than $x$ " is $x-10$ , not $10-x$ : take 10 away from $x$ .

2Terms and substituting

UseSubstitute in brackets; exponents before multiplying.

Example $2a-b^2$ at $a=5$ , $b=-3$ : $2(5)-(-3)^2=10-9=1$ .

Trap $-x^2$ at $x=-3$ is $-(-3)^2=-9$ , not $9$ : square first, then apply the minus.

TrapIn $10-2x$ : coefficient $-2$ , constant $10$ .

3Like terms

UseLike terms: same variables, same exponents. Add their coefficients; the variable part stays.

Example $6a^2+4a-a^2-9a=5a^2-5a$

Trap $3x^2y$ and $5xy^2$ are unlike: a different variable is squared.

Trap $5x+3y-x=4x+3y$ , not $8xy-x$ : unlike terms stay apart.

4Distributive property

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

$-a(b-c)=-ab+ac$

UseMultiply every term in the bracket by the number in front, sign included. Then collect like terms.

Example $4(x+3)-(2x-5)=4x+12-2x+5=2x+17$

Trap $-2(3x-4)=-6x+8$ , not $-6x-8$ : $(-2)(-4)=+8$ .

Trap $5-2(3x-1)=5-6x+2=7-6x$ . Doing $5-2=3$ first gives $9x-3$ : wrong.

5Solving equations

UseTwo steps: undo $+$ or $-$ , then $\times$ or $\div$ . Check in the original.

Undo with the inverse, on both sides:

$x+5=12$	subtract $5$ : $x=7$
$x-5=12$	add $5$ : $x=17$
$4x=10$	divide by $4$ : $x=2.5$
$\dfrac{x}{4}=10$	multiply by $4$ : $x=40$

Trap $2x+8=20$ : $2x=12$ , $x=6$ . Not $x+8=10$ : divide every term.

Trap $-x=7$ is not finished: $x=-7$ .

6Multi-step equations

Use1. Expand brackets. 2. Multiply every term by the LCD. 3. Collect $x$ terms on one side. 4. Divide.

Example $3(x-2)=x+8$ : $3x-6=x+8$ , $2x=14$ , $x=7$ . Check: $3(5)=15=7+8$ .

Trap $x/3+2=x/4$ , times $12$ : $4x+24=3x$ , not $4x+2=3x$ .

Trap $7x=4x+9$ : subtract $4x$ , so $3x=9$ , not $11x=9$ .

7Rearranging formulas

UseTreat the other letters as numbers. Undo the operations on your variable in reverse order, on both sides.

Example $F=1.8C+32$ for $C$ : $F-32=1.8C$ , so $C=(F-32)/1.8$ . At $F=86$ : $C=30$ .

TipCheck a formula with two sets of numbers, not one.

Trap $y=4x-10$ for $x$ : add $10$ first, so $x=(y+10)/4$ , not $y/4+10$ .

8Writing equations

$\text{consecutive: } x,\;x+1,\;x+2$

$\text{even or odd: } x,\;x+2,\;x+4$

UseLet $x$ be the unknown, with units. Is means $=$ .

ExampleThree consecutive odd numbers add to $87$ : $3x+6=87$ , $x=27$ : $27$ , $29$ , $31$ .

Trap4 times as many students as teachers: $s=4t$ , not $4s=t$ .

TrapAnswer with every number asked for, not only $x$ .

9Shapes, rates, money

$P=2l+2w$

$\text{average speed}=\dfrac{\text{total distance}}{\text{total time}}$

Use20% off, then 13% HST: multiply by $0.80$ , then by $1.13$ .

TrapPerimeter $100$ : $2w+2(w+10)=100$ , $w=20$ . Not $w+(w+10)=100$ : count all four sides.

Trap20% off, then 13% HST: $0.80\times1.13=0.904$ , so you pay 90.4%, not 93%.

Which move?

The equation has brackets Expand them

There are fractions Multiply by the LCD

x

is on both sides Collect

x

terms

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