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Open Math›Year 7›Algebraic techniques›Algebraic expressions

Algebraic expressions

Identify terms, coefficients and constants in algebraic expressions, and simplify expressions by collecting like terms.

Worked examples

Identifying parts of an algebraic expression

Straightforward

Problem

For the expression

5x - 3

, state the number of terms, the coefficient of

x

, and the constant term.

1

Count the terms. Terms are separated by

+

and

-

signs.

The terms are

5x

and

-3

.

There are

2

terms.

2

Identify the coefficient of

x

. This is the number multiplying

x

, including its sign.

The term containing

x

is

5x

.

Coefficient of

x

= 5

3

Identify the constant term. This is the term with no variable.

The term with no variable is

-3

.

Constant term

= -3

Answer

2 terms, coefficient of

x

is

5

, constant term is

-3

Collecting like terms

Moderate

Problem

Simplify

4x + 3y - 2x + y

.

1

Identify the like terms. Like terms have the same variable.

x

terms:

4x

and

-2x

y

terms:

3y

and

y

2

Collect the

x

terms.

4x - 2x = 2x

3

Collect the

y

terms. Note that

y

means

1y

.

3y + y = 3y + 1y = 4y

4

Write the simplified expression.

4x + 3y - 2x + y = 2x + 4y

Answer

2x + 4y

Expanding brackets and collecting like terms

Challenging

Problem

Expand and simplify

3(2x + 5) - 4x

.

1

Expand the brackets by multiplying each term inside by

3

.

3(2x + 5) = 3 \times 2x + 3 \times 5 = 6x + 15

2

Rewrite the full expression with the brackets expanded.

3(2x + 5) - 4x = 6x + 15 - 4x

3

Collect the like terms containing

x

.

6x - 4x = 2x

4

Write the simplified expression.

6x + 15 - 4x = 2x + 15

Answer

2x + 15

Practise

Q1·Straightforward

In the expression

3x + 5

, what is the coefficient of

x

?

$1$$3$$5$$x$

Explanation

In

3x + 5

, the term involving

x

is

3x

. The number multiplying

x

is

3

, so the coefficient of

x

is

3

.

Q2·Straightforward

In the expression

7y - 2

, what is the constant term?

$7$$y$$-2$$2$

Explanation

In

7y - 2

, the term

7y

contains the variable

y

, but

-2

has no variable. The constant term is

-2

.

Q3·Straightforward

How many terms are in the expression

4a + 3b - 6

?

$2$$3$$4$$6$

Explanation

The three terms are

4a

,

3b

, and

-6

. Terms are the parts of an expression separated by addition or subtraction.

Q4·Straightforward

In the expression

9 - 4m

, what is the coefficient of

m

?

$9$$-4$$4$$-9$

Explanation

The term involving

m

is

-4m

. The coefficient includes the sign, so the coefficient of

m

is

-4

.

Q5·Moderate

Simplify

3x + 5x

. What is the coefficient of

x

in the result?

Explanation

Both terms contain

x

, so they are like terms:

3x + 5x = (3 + 5)x = 8x

The coefficient of

x

is

8

.

Q6·Moderate

Simplify

7a - 3a + 2a

. What is the coefficient of

a

in the result?

Explanation

All three terms contain

a

:

7a - 3a + 2a = (7 - 3 + 2)a = 6a

The coefficient of

a

is

6

.

Q7·Moderate

Simplify

5m + 2n - 3m

. What is the coefficient of

m

in the simplified expression?

Explanation

The like terms containing

m

are

5m

and

-3m

. The term

2n

is different and stays separate:

5m + 2n - 3m = (5 - 3)m + 2n = 2m + 2n

The coefficient of

m

is

2

.

Q8·Moderate

Simplify

4p - p + 3q - 2q

. What is the coefficient of

q

in the simplified expression?

Explanation

Collect like terms:

4p - p + 3q - 2q = (4 - 1)p + (3 - 2)q = 3p + q

Note that

q

means

1q

, so the coefficient of

q

is

1

.

Q9·Challenging

Expand and simplify

3(x + 2) + x

. What is the coefficient of

x

in the result?

Explanation

Expand the brackets:

3(x + 2) = 3x + 6

Now collect like terms:

3x + 6 + x = (3 + 1)x + 6 = 4x + 6

The coefficient of

x

is

4

.

Q10·Challenging

Expand and simplify

2(a + 3b) - a

. What is the coefficient of

a

in the result?

Explanation

Expand the brackets:

2(a + 3b) = 2a + 6b

Now collect like terms:

2a + 6b - a = (2 - 1)a + 6b = a + 6b

The coefficient of

a

is

1

.

Q11·Challenging

Expand and simplify

4(2x - y) + 3y

. What is the coefficient of

y

in the result?

Explanation

Expand the brackets:

4(2x - y) = 8x - 4y

Now collect like terms:

8x - 4y + 3y = 8x + (-4 + 3)y = 8x - y

The coefficient of

y

is

-1

.

Q12·Challenging

Expand and simplify

5x + 2(x - 3y) - 4y

. What is the coefficient of

y

in the result?

Explanation

Expand the brackets:

2(x - 3y) = 2x - 6y

Rewrite the full expression:

5x + 2x - 6y - 4y

Collect like terms:

= (5 + 2)x + (-6 - 4)y = 7x - 10y

The coefficient of

y

is

-10

.

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