Algebraic expressions

Q: What is the difference between an expression and an equation?

An expression has no equals sign (2x + 3). An equation says two expressions are equal (2x + 3 = 11) and can be solved.

Q: What is a coefficient?

The number multiplying a variable. In 7y the coefficient is 7; in -x it is -1.

Algebra uses letters, called variables, to stand for numbers. An algebraic expression such as $3 n + 5$ combines variables and numbers with operations. Expressions describe patterns and rules, and they can be simplified and evaluated.

Formulas

Parts of an expression: in 4x + 7, 4 is the coefficient, x the variable and 7 the constant

4 x + 7

Only like terms can be added or subtracted

3 x + 5 x = 8 x 3 x + 5 y stays as it is

Finding the value: substitute, then calculate

x = 3 : 2 x + 1 = 2 \times 3 + 1 = 7

Worked examples

Example 1. Write an expression for "5 more than 3 times a number

n

3 times the number is $3 n$ .
5 more than that is $3 n + 5$ .

Answer:

3 n + 5

Example 2. Simplify

4 a + 3 b - a + 2 b

Group like terms: $(4 a - a) + (3 b + 2 b)$ .
$= 3 a + 5 b$ .

Answer:

3 a + 5 b

Example 3. Find the value of

2 x^{2} - 3 x + 1

when

x = 3

$2 \times 3^{2} - 3 \times 3 + 1 = 2 \times 9 - 9 + 1$ .
$= 18 - 9 + 1 = 10$ .

Answer: 10

Example 4. A row of squares is made from matchsticks: 1 square needs 4 sticks, and each new square needs 3 more. How many sticks for

n

squares, and for 10 squares?

Sticks $= 4 + 3 (n - 1) = 3 n + 1$ .
For $n = 10 : 3 \times 10 + 1 = 31$ .

Answer:

3 n + 1

; 31 sticks

Common mistakes

Adding unlike terms: $3 x + 2 y$ is not $5 x y$ . It cannot be simplified.
Mixing up $3 n$ (3 times $n$ ) and $n + 3$ (3 more than $n$ ).
Sign errors when substituting negative numbers. Use brackets: for $x = - 2, x^{2} = (- 2)^{2} = 4$ .

Practice questions with answers

1. What does

6 n

mean?

A6 plus n
B6 times n
Cn more than 6
Dn divided by 6

Show answer

Answer: B) 6 times n

When a number is next to a variable with no operator between them, it means multiplication. So

6 n = 6 \times n

2. Which of the following is a variable?

A7
B $x$
C12
D100

Show answer

Answer: B)

x

A variable is a letter (like

x

) that can take different values. 7, 12 and 100 are fixed numbers (constants).

3. Which expression means '5 more than

x

A $x + 5$
B $x - 5$
C $5 x$
D $\frac{x}{5}$

Show answer

Answer: A)

x + 5

'More than' means addition, so 5 more than

x

x + 5

4. Look at the pattern: 2, 4, 6, 8, ... What is the next number?

Show answer

Answer: B) 10

The numbers increase by 2 each time, so after 8 comes 8 + 2 = 10.

5. In the rule P = 4l for a square, what does the letter l stand for?

AThe number of sides
BThe area
CThe length of one side
DThe perimeter

Show answer

Answer: C) The length of one side

l is the variable for the side length. Multiplying it by 4 (the number of sides) gives the perimeter P.

6. What is the coefficient in the term

7 x

A7
B $x$
C1
D $7 x$

Show answer

Answer: A) 7

The coefficient is the numerical part of the term, which is 7.

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Frequently asked questions

What is a variable?

A letter that stands for a number that can change or is not known yet, such as $n$ in " $n$ pencils".

What are like terms?

Terms with exactly the same variables raised to the same powers: $5 x$ and $- 2 x$ are like terms; $5 x$ and $5 x^{2}$ are not.

What is the difference between an expression and an equation?

An expression has no equals sign $(2 x + 3)$ . An equation says two expressions are equal $(2 x + 3 = 11)$ and can be solved.

What is a coefficient?

The number multiplying a variable. In $7 y$ the coefficient is 7; in $- x$ it is $- 1$ .

Algebraic expressions

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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