Algebraic expressions: like terms and substitution – Linear Functions, Equations and Inequalities (Senior 1 Mathematics) | Ganzaa

Linear Functions, Equations and Inequalities

Lessons

Senior 1 · Mathematics

Linear Functions, Equations and Inequalities

5 lessons

Lesson 3 of 5

Algebraic expressions: like terms and substitution

22 min

Let's begin

At a school canteen a mandazi costs $m$ francs and a cup of tea costs $t$ francs. Uwase buys 3 mandazi and 2 cups of tea, so she pays $3 m + 2 t$ . Her friend Gatete buys 2 mandazi and 1 tea: $2 m + t$ . Together they pay $5 m + 3 t$ . Mandazi can be added to mandazi and tea to tea, but mandazi cannot be added to tea. That one idea is the whole of this lesson.

1. The vocabulary of algebra

Definition

Variable: a letter that stands for a number which is unknown or can change, such as $m$ or $x$ .

Algebraic expression: numbers and variables joined by operation signs, with no equals sign. Example: $3 m + 2 t - 5$ .

Definition

Term: one part of an expression, separated from the others by $+$ or $-$ . The sign in front belongs to the term. Example: $3 m + 2 t - 5$ has the three terms $3 m$ , $2 t$ and $- 5$ .

Coefficient: the number that multiplies the variable in a term (3 in $3 m$ ; $- 1$ in $- x$ ). Constant: a term with no variable ( $- 5$ above).

Definition

Like terms: terms with exactly the same variables, each raised to the same power. Only the coefficients may differ. Unlike terms differ in a variable or a power.

Examples: $3 m$ and $- 7 m$ are like terms; $2 x^{2} y$ and $5 x^{2} y$ are like terms; $3 m$ and $2 t$ are unlike; $x$ and $x^{2}$ are unlike.

2. Notation

You write	It means	Note
$3 m$	$3 \times m$	the number comes first, and the $\times$ sign is left out
$m n$	$m \times n$	$m n$ and $n m$ are the same term
$m^{2}$	$m \times m$	not the same as $2 m = m + m$
$\frac{m}{4}$	$m \div 4$	division is written as a fraction
$m$ , $- m$	$1 m$ , $- 1 m$	a coefficient of 1 is not written

3. Rule: collecting like terms

To simplify, add or subtract the coefficients of like terms and keep the variable part unchanged. Unlike terms are left as they are.

Why it works. By the distributive law, $3 m + 2 m = (3 + 2) m = 5 m$ : three lots of $m$ and two lots of $m$ make five lots of $m$ . No law turns $3 m + 2 t$ into one term, because $m$ and $t$ can be different numbers.

Worked example

Simplify (a) $4 x + 3 y - x + 5 y - 2$ ; (b) $2 a^{2} + 3 a - a^{2} + 4 a - 7$

(a) Group the like terms, each carrying the sign in front of it: $(4 x - x) + (3 y + 5 y) - 2 = 3 x + 8 y - 2.$

(b) $a^{2}$ and $a$ are unlike, so they are grouped separately: $(2 a^{2} - a^{2}) + (3 a + 4 a) - 7 = a^{2} + 7 a - 7.$

Worked example

Simplify $3 (2 p - q) - 2 (p + 4 q)$

Multiply out each bracket. The $- 2$ multiplies both terms in the second bracket: $6 p - 3 q - 2 p - 8 q$ .
Collect like terms: $(6 p - 2 p) + (- 3 q - 8 q) = 4 p - 11 q$ .

4. Substitution

Definition

Substitution: replacing each variable in an expression by a given number, then working out the value.

Example: when $m = 150$ and $t = 200$ , Uwase's bill $3 m + 2 t = 3 (150) + 2 (200) = 450 + 400 = 850$ Frw.

Important

How to substitute. Write each value in brackets, then follow the order of operations: brackets, then powers, then $\times$ and $\div$ from left to right, then $+$ and $-$ from left to right. The brackets keep the sign of a negative number with the number.

Worked example

Given $a = 3$ , $b = - 2$ and $c = 4$ , evaluate each expression

Expression	Substitute	Value
$2 a + 3 b - c$	$2 (3) + 3 (- 2) - 4 = 6 - 6 - 4$	$- 4$
$a b^{2}$	$3 \times (-2)^2 = 3 \times 4$	$12$
$5 a b c$	$5 (3) (- 2) (4)$	$- 120$
$\dfrac{a + c}{b}$	$\dfrac{3 + 4}{-2}$	$- 3.5$
$\dfrac{4c - a}{a + b}$	$\dfrac{16 - 3}{3 + (-2)} = \dfrac{13}{1}$	$13$

In $a b^{2}$ only $b$ is squared, because a power acts before a multiplication.

Worked example

Suppose a daily data bundle costs 200 Frw and a weekly bundle costs 1,000 Frw. Write the cost $C$ of $d$ daily and $w$ weekly bundles, then find $C$ when $d = 4$ and $w = 2$

C = 200 d + 1,000 w

C = 200(4) + 1{,}000(2) = 800 + 2{,}000 = 2{,}800 \text{ Frw}.

Warning

$(- 2)^{2}$ is not $- 2^{2}$ . If $b = - 2$ , then $b^{2} = (- 2)^{2} = (- 2) (- 2) = 4$ . Writing $- 2^{2}$ without brackets means $- (2^{2}) = - 4$ . Always put a substituted negative number in brackets.

Warning

$3 m + 2 t$ is not $5 m t$ . Unlike terms cannot be added. Test it: with $m = 1$ and $t = 1$ , $3 m + 2 t = 5$ and $5 m t = 5$ too, which proves nothing; but with $m = 2$ and $t = 1$ , $3 m + 2 t = 8$ while $5 m t = 10$ . One example that fails is enough to show the step is wrong.

Tip

Before you collect terms, mark each kind in its own way: circle the $x$ -terms, underline the $y$ -terms, and box the numbers, each together with the sign in front of it. Then add up one kind at a time.

Khan Academy: "Combining like terms introduction". Watch how the terms are grouped by their variable before the coefficients are added.

✦

Practice

Show each step.

List the terms of $5 x - 3 y + 7$ and give the coefficient of $y$ .
Which of these are like terms: $4 a b$ , $3 a$ , $- a b$ , $2 b$ , $7 b a$ ?
Simplify $7 k + 2 k - 4 k$ .
Simplify $6 m - 2 n + 3 m + 5 n - 1$ .
Simplify $3 x^{2} - 5 x + x^{2} + 2 x$ .
Given $p = 4$ and $q = - 3$ , evaluate (a) $3 p - 2 q$ ; (b) $p q^{2}$ ; (c) $\frac{p - q}{7}$ .
Simplify $2 (3 a + b) - (a - 4 b)$ .
Beans cost 900 Frw per kg and maize flour costs 700 Frw per kg. Write an expression for the cost of $x$ kg of beans and $y$ kg of flour, then find the cost when $x = 5$ and $y = 3$ .

Show answersHide answers

$5 x$ , $- 3 y$ , $7$ . The coefficient of $y$ is $- 3$ .
$4 a b$ , $- a b$ and $7 b a$ ( $b a$ is the same as $a b$ ).
$5 k$
$9 m + 3 n - 1$
$4 x^{2} - 3 x$
(a) $12 + 6 = 18$ ; (b) $4 \times 9 = 36$ ; (c) $\frac{7}{7} = 1$ .
$6 a + 2 b - a + 4 b = 5 a + 6 b$
$900 x + 700 y$ ; $900 (5) + 700 (3) = 4,500 + 2,100 = 6,600$ Frw.

Sources and credits

1.3: Use the Language of Algebra, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Combining like terms introduction, Khan Academy, embedded from YouTube.

The worked examples, diagrams, practice items and quiz questions were written by Ganzaa.

11. The vocabulary of algebra

22. Notation

33. Rule: collecting like terms

44. Substitution

1. The vocabulary of algebra

2. Notation

3. Rule: collecting like terms

4. Substitution