Algebraic Expressions and Equations: Test Review

Everything for the Unit 3 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Writing Expressions from Patterns and Words

A variable stands for a number that can change; a coefficient multiplies a variable; a constant is a term with no variable.
An expression has no equals sign ( $3 n + 2$ ); an equation says two things are equal ( $3 n + 2 = 20$ ).
The sign in front of a term belongs to it: in $4 - 3 x$ the coefficient of $x$ is $- 3$ . The coefficient of $x$ alone is $1$ , and of $- x$ is $- 1$ .
For a pattern that grows by the same amount each stage: expression = (change) $n$ + (stage 0 value), where stage 0 value = stage 1 value − change.
"5 less than $n$ " is $n - 5$ ; "twice $n$ " is $2 n$ ; "the square of $n$ " is $n^{2}$ ; "triple the sum of $n$ and 6" is $3 (n + 6)$ .

Equivalent Expressions and Simplifying

Like terms have the same variables with the same exponents; collect them by adding coefficients: $3 x + 5 x = 8 x$ .
Distributive property: $a (b + c) = ab + a c$ . Multiply every term inside the brackets.
A negative outside brackets changes every sign inside: $- 2 (x - 4) = - 2 x + 8$ and $- (x + 3) = - x - 3$ .
Two expressions are equivalent if they are equal for every value of the variable. One value where they differ proves they are not equivalent.
A zero pair ( $+ 1$ and $- 1$ ) has value $0$ and can be removed.

Solving Linear Equations

Whatever you do to one side of an equation, do to the other side (and to every term on that side).
Undo operations in reverse order: for $3 x + 2 = 11$ , subtract 2 first, then divide by 3.
With variables on both sides, gather the $x$ terms on one side and the constants on the other.
Clear fractions by multiplying every term on both sides by the lowest common denominator; keep each numerator in brackets.
Verify by substituting into the original equation: the left side and the right side must give the same number.
If the $x$ terms cancel: a false statement (like $3 = 5$ ) means no solution; a true statement (like $2 = 2$ ) means every $x$ is a solution.

Equations in Context and Rearranging Formulas

Word problems: define the variable (with units), write the equation, solve, check in the situation, answer in a sentence.
To rearrange a formula, use the same inverse operations as solving an equation, treating the other letters like numbers.
Undo in reverse order: $P = 2 l + 2 w$ becomes $w = \frac{P - 2 l}{2}$ (subtract $2 l$ , then divide by 2).
$d = s t$ rearranges to $t = \frac{d}{s}$ and $s = \frac{d}{t}$ ; $C = 2 π r$ rearranges to $r = \frac{C}{2 π}$ .
A rearranged formula with a variable in the denominator only works when that variable is not zero.

Where marks get lost

Writing "5 less than a number" as $5 - n$ . The number comes first: $n - 5$ . (Writing Expressions from Patterns and Words)
Using the stage 1 value as the constant. For the pattern 5, 8, 11 the expression is $3 n + 2$ , not $3 n + 5$ : test $n = 1$ and $3 (1) + 5 = 8$ , which is wrong. (Writing Expressions from Patterns and Words)
Dropping the sign of a term: in $4 - 3 x$ the coefficient of $x$ is $- 3$ , not $3$ . (Writing Expressions from Patterns and Words)
Saying $x$ has no coefficient. It has coefficient $1$ , and $- x$ has coefficient $- 1$ . (Writing Expressions from Patterns and Words)
Mixing up "twice a number" ( $2 n$ ) and "the square of a number" ( $n^{2}$ ), or leaving out brackets: "triple the sum of $n$ and 6" is $3 (n + 6)$ , not $3 n + 6$ . (Writing Expressions from Patterns and Words)
Combining unlike terms: writing $3 x + 2 = 5 x$ , or $x + x^{2} = 2 x^{2}$ . Only terms with the same variable part combine. (Equivalent Expressions and Simplifying)
Changing the exponent when adding: $3 x + 5 x = 8 x$ , not $8 x^{2}$ . Adding like terms changes the coefficient only. (Equivalent Expressions and Simplifying)
Distributing to the first term only: $3 (x + 4) = 3 x + 4$ is wrong; it is $3 x + 12$ . (Equivalent Expressions and Simplifying)
Losing a sign with a negative outside: $- 2 (x - 4)$ is $- 2 x + 8$ , not $- 2 x - 8$ , because $(- 2) (- 4) = + 8$ . (Equivalent Expressions and Simplifying)
Deciding two expressions are equivalent because they match at one value (often $x = 0$ or $x = 1$ ). One match proves nothing; one mismatch proves they are not equivalent. (Equivalent Expressions and Simplifying)
Doing an operation to one side only, or to only some terms: multiplying $\frac{x}{4} + 3 = 7$ by 4 and writing $x + 3 = 28$ instead of $x + 12 = 28$ . (Solving Linear Equations)
Losing the sign of the coefficient: from $- 4 x = 16$ writing $x = 4$ . Divide by $- 4$ , so $x = - 4$ . (Solving Linear Equations)
Moving a term across the equals sign without changing the operation: from $5 x = 2 x + 11$ writing $5 x + 2 x = 11$ . Subtracting $2 x$ from both sides gives $3 x = 11$ . (Solving Linear Equations)
Forgetting brackets around a numerator: multiplying $\frac{x - 1}{2}$ by 6 gives $3 (x - 1) = 3 x - 3$ , not $3 x - 1$ . (Solving Linear Equations)
Checking the answer in a later line instead of the original equation, so an early mistake is never caught. (Solving Linear Equations)
Dividing only part of a side: from $P - 2 l = 2 w$ writing $w = P - \frac{2 l}{2} = P - l$ . Divide the whole side by 2: $w = \frac{P - 2 l}{2}$ . (Equations in Context and Rearranging Formulas)
Undoing in the wrong order: from $F = 1.8 C + 32$ writing $C = \frac{F}{1.8} - 32$ . The 32 was added last, so subtract it first: $C = \frac{F - 32}{1.8}$ . (Equations in Context and Rearranging Formulas)
Flipping a quotient: from $d = s t$ writing $t = \frac{s}{d}$ . Check with numbers: 120 km at 60 km/h takes $\frac{120}{60} = 2$ hours, so $t = \frac{d}{s}$ . (Equations in Context and Rearranging Formulas)
Not defining the variable, then mixing units, such as using a time in minutes with a speed in km/h. (Equations in Context and Rearranging Formulas)
Giving an answer that fits the equation but not the situation, like 12.4 tickets or a negative length. (Equations in Context and Rearranging Formulas)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Writing Expressions from Patterns and Words
1
What is the coefficient of $y$ in $7 x - y + 4$ ?
Writing Expressions from Patterns and Words
2
A tile pattern has 7 tiles in stage 1, 12 in stage 2 and 17 in stage 3. If it keeps growing the same way, how many tiles are in stage 30?
tiles
Writing Expressions from Patterns and Words
3
Evaluate $3 (n + 4) - n$ when $n = 6$ .
Writing Expressions from Patterns and Words
4
Which of these is an equation rather than an expression?
Writing Expressions from Patterns and Words
5
The word algebra comes from al-jabr, a word in the title of a book written around 820 CE by which scholar?
Writing Expressions from Patterns and Words
6
A phone repair shop charges a 30 dollar inspection fee plus 45 dollars per hour of work, so the cost is $30 + 45 h$ . What is the cost, in dollars, for 3.5 hours of work?
dollars
Equivalent Expressions and Simplifying
7
Simplify $4 x + 7 - x + 2$ .
Equivalent Expressions and Simplifying
8
Expand $- 3 (x - 4)$ .
Equivalent Expressions and Simplifying
9
Which expression is equivalent to $4 (n + 1)$ ?
Equivalent Expressions and Simplifying
10
Match each expression to its simplified form.
1. A. $- x + 6$
2. B. $3 x + 6$
3. C. $3 x$
4. D. $6$
5. E. $x^{3}$
$3 (x + 2)$
$- (x - 6)$
$x + x + x$
$x \cdot x \cdot x$
$2 x + 6 - 2 x$
Equivalent Expressions and Simplifying
11
A square pool is 12 m by 12 m. How many 1 m square tiles are needed for a border one tile wide all the way around it?
tiles
Equivalent Expressions and Simplifying
12
What is $3 x + 5 x$ ?
Solving Linear Equations
13
Solve $4 x - 7 = 21$ .
Solving Linear Equations
14
Solve $6 x + 4 = 2 x - 12$ .
Solving Linear Equations
15
Solve $\frac{x}{5} - 2 = 4$ .
Solving Linear Equations
16
To solve $3 x + 2 = 11$ without creating fractions, what should you do first?
Solving Linear Equations
17
Put these steps in order to solve (x + 2)/3 = (x − 1)/2.
1. 1Expand: 2x + 4 = 3x − 3
2. 2Check: (7 + 2)/3 = 3 and (7 − 1)/2 = 3
3. 3Collect x terms on one side and constants on the other: 7 = x
4. 4Multiply both sides by 6: 2(x + 2) = 3(x − 1)
Solving Linear Equations
18
Is $x = 4$ a solution of $5 x - 3 = 2 x + 9$ ? Answer yes or no.
Equations in Context and Rearranging Formulas
19
A moving truck rents for 55 dollars a day plus 0.25 dollars per kilometre. With a budget of 105 dollars for one day, how many kilometres can you drive?
km
Equations in Context and Rearranging Formulas
20
A rectangle's length is twice its width, and its perimeter is 54 cm. What is its width?
cm
Equations in Context and Rearranging Formulas
21
Rearrange $P = 2 l + 2 w$ to make $l$ the subject.
Equations in Context and Rearranging Formulas
22
A circle has a circumference of 100 cm. Use $r = \frac{C}{2 π}$ to find its radius, rounded to one decimal place.
cm
Equations in Context and Rearranging Formulas
23
A taxi charges a 4 dollar base fare plus 2 dollars per kilometre, and a ride costs 22 dollars. Which equation models the ride, with k as the distance in kilometres?
Equations in Context and Rearranging Formulas
24
Match each formula with the rearrangement asked for.
1. A. $r = \frac{C}{2 π}$
2. B. $s = \frac{d}{t}$
3. C. $s = \frac{P}{4}$
4. D. $w = \frac{A}{l}$
$A = lw$ , solved for $w$
$d = s t$ , solved for $s$
$P = 4 s$ , solved for $s$
$C = 2 π r$ , solved for $r$

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…