MTH1W Equivalent Expressions and Simplifying: Notes and Practice

A square pool measures $n$ metres by $n$ metres, and a border of 1-metre square tiles goes all the way around it. Three friends write a rule for the number of border tiles. Aisha counts the four sides and then the four corners: $4 n + 4$ . Ben splits the border into four equal strips, each $n + 1$ tiles long: $4 (n + 1)$ . Chen counts the top and bottom rows of $n + 2$ tiles, then the two sides of $n$ tiles: $2 (n + 2) + 2 n$ . They look different, but try a 5 m pool: $4 (5) + 4 = 24$ , $4 (6) = 24$ and $2 (7) + 2 (5) = 24$ . Expressions that give the same value for every value of the variable are called equivalent expressions. This lesson shows you how to simplify expressions and how to prove (or disprove) that two of them are equivalent.

Like terms

Like terms have exactly the same variable part: the same letters raised to the same exponents. Only the coefficients can differ. So $3 x$ and $- 5 x$ are like terms, $x^{2}$ and $4 x^{2}$ are like terms, and all constants (like $7$ and $- 2$ ) are like terms. But $x$ and $x^{2}$ are not alike, and neither are $x$ and $y$ . To collect like terms, add their coefficients and keep the variable part the same. The reason is the same as counting objects: 3 apples plus 5 apples is 8 apples, not 8 "apples squared". In the same way, $3 x + 5 x = 8 x$ . Unlike terms can't be combined into one term: $3 x + 2$ stays as it is, just as 3 apples plus 2 oranges is not 5 of anything.

Pair	Like terms?	Why
$3 x$ and $- 5 x$	yes	same variable, x
$x^{2}$ and $4 x^{2}$	yes	same variable and same exponent
$2 x y$ and $- y x$	yes	xy and yx are the same product
$7$ and $- 2$	yes	both constants
$x$ and $x^{2}$	no	different exponents
$3 x$ and $3 y$	no	different variables
$4 x$ and $4$	no	one has a variable, one does not

Fig 1Algebra tiles. Left: two groups of x + 3 regroup into 2x + 6. Right: (3x + 2) + (x − 5) collects to 4x − 3 after two zero pairs cancel.

Algebra tiles make collecting like terms visible. A long tile stands for $x$ , a small square stands for $1$ , and a tile in the negative colour stands for $- 1$ . One positive unit tile and one negative unit tile together make $0$ , a zero pair, so you can remove them without changing the value. In the right half of the figure, $(3 x + 2) + (x - 5)$ becomes 4 long tiles, plus 2 positive and 5 negative unit tiles. Two zero pairs cancel and 3 negative tiles are left: $4 x - 3$ . Grouping like terms in writing does the same job: $3 x + x = 4 x$ and $2 - 5 = - 3$ .

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Which pair are like terms?

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Simplify

5 a + 3 - 2 a - 7

VideoJensenMath, an Ontario teacher, shows how to spot like terms and collect them, keeping each term's sign attached, in a Grade 9 lesson.Watch on YouTube

The distributive property

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

Multiply the number outside the brackets by every term inside.

The distributive property says that multiplying a sum by a number is the same as multiplying each part and then adding. In the left half of the figure, $2 (x + 3)$ means two groups of $x + 3$ , which is 2 long tiles and 6 unit tiles: $2 x + 6$ . Every term inside the brackets gets multiplied, not just the first one. Watch the signs when the number outside is negative: $- 2 (x - 4) = (- 2) (x) + (- 2) (- 4) = - 2 x + 8,$ because a negative times a negative is positive. A minus sign in front of brackets means multiply by $- 1$ : $- (x + 3) = - x - 3$ . Writing the expression without brackets is called expanding.

Fig 2Left: a rectangle 3 tall and x + 4 wide splits into areas 3x and 12, so 3(x + 4) = 3x + 12. Right: −2 multiplies both x and −4, giving −2x + 8.

3 (2 x - 1) - 2 (x + 4) = 6 x - 3 - 2 x - 8 = 4 x - 11

Expand both brackets (the second with −2), then collect like terms.

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Expand

- (x - 7)

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Expand

4 (2 x + 3)

VideoMath Antics explains the distributive property: the number outside the brackets multiplies every term inside, not just the first one.Watch on YouTube

Checking whether two expressions are equivalent

There are four ways to check. Algebra: simplify both expressions; if they simplify to the same thing, they are equivalent. Aisha's, Ben's and Chen's rules all simplify to $4 n + 4$ . Substitution: put the same value into both. If the results differ for even one value, the expressions are not equivalent; one counterexample settles it. A match at one value does not prove equivalence, though: $2 x$ and $x^{2}$ both give $4$ when $x = 2$ , but at $x = 3$ they give $6$ and $9$ . For linear expressions in one variable (no exponents above 1), matching at two different values is enough, because their graphs are straight lines and two points fix a line. Graphs: equivalent expressions graph as the exact same line or curve. Tiles: equivalent expressions can be built from the same set of tiles once zero pairs are removed.

Fig 3Left: every point of 4(n + 1) lies on the line of 4n + 4, while the dashed line 4n is 4 lower everywhere. Right: the line 2x and the curve x² meet at x = 0 and x = 2 but give 6 and 9 at x = 3.

Pool side n (m)	4n + 4	4(n + 1)	2(n + 2) + 2n	4n (missing corners)
1	8	8	8	4
2	12	12	12	8
5	24	24	24	20
10	44	44	44	40

Three equivalent border rules agree for every pool; the rule 4n is always 4 short because it misses the corner tiles.

AEquivalent

$2 (x + 3)$ and $2 x + 6$
$5 x - 2 x + 1$ and $3 x + 1$
$- (x - 6)$ and $6 - x$
$x + x + x$ and $3 x$

BNot equivalent

$2 (x + 3)$ and $2 x + 3$ (try $x = 0$ : 6 vs 3)
$3 x + 2 x$ and $5 x^{2}$ (try $x = 2$ : 10 vs 20)
$x \cdot x \cdot x$ and $3 x$ (try $x = 2$ : 8 vs 6)
$n - 5$ and $5 - n$ (try $n = 8$ : 3 vs −3)

Pick a test value that can catch a difference

Values like

0

and

1

hide differences:

x

x^{2}

and

x^{3}

are all equal at

x = 0

and again at

x = 1

. Test with something like

x = 3

x = - 2

instead. If the values match, simplify algebraically to be sure.

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Which test value would not reveal that

x^{2}

and

x^{3}

are different?

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Two linear expressions in

x

give the same value at

x = 2

and again at

x = 5

. What can you conclude?

Equivalent Expressions and Simplifying

Key things to remember

Like terms

The distributive property

Checking whether two expressions are equivalent

Worked examples

Where marks get lost

Flashcards 8

Practice