Sec 1 · G3 (Express)

Linear equations

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1. Solving linear equations

To undo	Do to both sides
$+ 5$	$- 5$
$- 5$	$+ 5$
$\times 3$	$\div 3$
$\div 3$	$\times 3$

Undo + 5, then × 3

\displaystyle 3x + 5 = 17

e.g.

\displaystyle 3x + 5 = 17

\displaystyle 3x = 12

\displaystyle x = 4

Letters on both sides: collect them on one side

\displaystyle 7x - 4 = 3x + 8

e.g.

\displaystyle 7x - 4 = 3x + 8

\displaystyle 4x = 12

\displaystyle x = 3

Expand the brackets first

\displaystyle 2(x - 3) = 4x + 1

e.g.

\displaystyle 2(x - 3) = 4x + 1

\displaystyle 2x - 6 = 4x + 1

\displaystyle -7 = 2x

\displaystyle x = -\dfrac{7}{2}

📖 Worked example

Solve $9x + 2 = 4x + 12$ .

Step 1 of 3

Collect the x terms on the left and the numbers on the right (a term changes sign as it crosses)

9x - 4x = 12 - 2

📖 Worked example

Solve $-3(3x + 9) = 9$ .

Step 1 of 4

Expand the brackets

-9x - 27 = 9

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2. Equations with fractions

Multiply both sides by the LCM of the denominators

\displaystyle \dfrac{x + 1}{3} = \dfrac{x - 2}{2}

e.g.

\displaystyle \dfrac{x + 1}{3} = \dfrac{x - 2}{2}

\displaystyle 2(x + 1) = 3(x - 2)

\displaystyle 2x + 2 = 3x - 6

\displaystyle x = 8

Every term is multiplied, including whole numbers

\displaystyle \dfrac{x}{4} + 2 = \dfrac{x}{3}

e.g.

\displaystyle \dfrac{x}{4} + 2 = \dfrac{x}{3}

\displaystyle 3x + 24 = 4x

\displaystyle x = 24

📖 Worked example

Solve $\dfrac{4x + 2}{6} = \dfrac{x - 1}{3}$ .

Step 1 of 5

Cross-multiply: each top times the other side's bottom

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

📖 Worked example

Given that $x = -1$ is the solution of the equation $\dfrac{x - 2k + 7}{6} = \dfrac{3x + 9}{3}$ , find the value of the constant $k$ .

Step 1 of 6

Cross-multiply: each top times the other side's bottom

3(x - 2k + 7) = 6(3x + 9)

Practise this →

3. Word problems

Step	Example: Ali is 4 years older than Ben; together they are 30
let the unknown be $x$	Ben is $x$ years old
form an equation	$x + (x + 4) = 30$
solve	$x = 13$
answer in words	Ben is $13$ , Ali is $17$

📖 Worked example

Aisyah's sister is $5$ years older than Aisyah. In $7$ years' time, the sum of their ages will be $37$ years. Find Aisyah's present age.

Step 1 of 3

Let her present age be x years. Their ages in the future

x + 7

x + 12

Two-digit number	Tens digit $t$ , ones digit $u$
the number	$10t + u$
digits reversed	$10u + t$
e.g. $t = 7$ , $u = 5$	$75$ and $57$

📖 Worked example

The tens digit of a two-digit number is $2$ more than its ones digit. When the digits are reversed, the new number and the original number add up to $132$ . Let the ones digit be $x$ . Form an equation in $x$ , and hence find the original number.

Step 1 of 6

The ones digit is x. The tens digit is

x + 2

📖 Worked example

On Monday, a pump filled an empty tank at a rate of $6$ litres per minute. On Tuesday, a smaller pump filled the same empty tank at $5$ litres per minute and took $3$ minutes longer. Let the time taken on Monday be $x$ minutes. Form an equation in $x$ , and hence find the capacity of the tank, in litres.

Step 1 of 5

Amount = rate × time. In litres, the first time, and the second time (d minutes longer)

6x

5(x + 3)

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4. Formulae

Substitute into a formula: P = 2(l + b), l = 8, b = 5

\displaystyle P = 2(l + b)

e.g.

\displaystyle 2(8 + 5)

\displaystyle = 26

📖 Worked example

The formula $v = u + at$ gives the speed of a moving object. Find the value of $t$ when $v = 18$ , $u = 2$ and $a = 4$ .

Step 1 of 3

Substitute the values you know

18 = 2 + 4t

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5. Preview (Sec 2): changing the subject

Make x the subject: undo in reverse order

\displaystyle y = 3x - 7

e.g.

\displaystyle y + 7 = 3x

\displaystyle x = \dfrac{y + 7}{3}

Make b the subject

\displaystyle P = 2(l + b)

e.g.

\displaystyle \dfrac{P}{2} = l + b

\displaystyle b = \dfrac{P}{2} - l

📖 Worked example

Make $x$ the subject of $y = -3x - 3$ .

Step 1 of 2

Get the x term on its own

y + 3 = -3x

Practise this →

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Sec 1 · G3 (Express)

Linear equations

Practise linear equations with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Solving linear equations

To undo	Do to both sides
$+ 5$	$- 5$
$- 5$	$+ 5$
$\times 3$	$\div 3$
$\div 3$	$\times 3$

Undo + 5, then × 3

\displaystyle 3x + 5 = 17

e.g.

\displaystyle 3x + 5 = 17

\displaystyle 3x = 12

\displaystyle x = 4

Letters on both sides: collect them on one side

\displaystyle 7x - 4 = 3x + 8

e.g.

\displaystyle 7x - 4 = 3x + 8

\displaystyle 4x = 12

\displaystyle x = 3

Expand the brackets first

\displaystyle 2(x - 3) = 4x + 1

e.g.

\displaystyle 2(x - 3) = 4x + 1

\displaystyle 2x - 6 = 4x + 1

\displaystyle -7 = 2x

\displaystyle x = -\dfrac{7}{2}

📖 Worked example

Solve $9x + 2 = 4x + 12$ .

Step 1 of 3

Collect the x terms on the left and the numbers on the right (a term changes sign as it crosses)

9x - 4x = 12 - 2

📖 Worked example

Solve $-3(3x + 9) = 9$ .

Step 1 of 4

Expand the brackets

-9x - 27 = 9

Practise this →

2. Equations with fractions

Multiply both sides by the LCM of the denominators

\displaystyle \dfrac{x + 1}{3} = \dfrac{x - 2}{2}

e.g.

\displaystyle \dfrac{x + 1}{3} = \dfrac{x - 2}{2}

\displaystyle 2(x + 1) = 3(x - 2)

\displaystyle 2x + 2 = 3x - 6

\displaystyle x = 8

Every term is multiplied, including whole numbers

\displaystyle \dfrac{x}{4} + 2 = \dfrac{x}{3}

e.g.

\displaystyle \dfrac{x}{4} + 2 = \dfrac{x}{3}

\displaystyle 3x + 24 = 4x

\displaystyle x = 24

📖 Worked example

Solve $\dfrac{4x + 2}{6} = \dfrac{x - 1}{3}$ .

Step 1 of 5

Cross-multiply: each top times the other side's bottom

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

📖 Worked example

Given that $x = -1$ is the solution of the equation $\dfrac{x - 2k + 7}{6} = \dfrac{3x + 9}{3}$ , find the value of the constant $k$ .

Step 1 of 6

Cross-multiply: each top times the other side's bottom

3(x - 2k + 7) = 6(3x + 9)

Practise this →

3. Word problems

Step	Example: Ali is 4 years older than Ben; together they are 30
let the unknown be $x$	Ben is $x$ years old
form an equation	$x + (x + 4) = 30$
solve	$x = 13$
answer in words	Ben is $13$ , Ali is $17$

📖 Worked example

Aisyah's sister is $5$ years older than Aisyah. In $7$ years' time, the sum of their ages will be $37$ years. Find Aisyah's present age.

Step 1 of 3

Let her present age be x years. Their ages in the future

x + 7

x + 12

Two-digit number	Tens digit $t$ , ones digit $u$
the number	$10t + u$
digits reversed	$10u + t$
e.g. $t = 7$ , $u = 5$	$75$ and $57$

📖 Worked example

Step 1 of 6

The ones digit is x. The tens digit is

x + 2

📖 Worked example

Step 1 of 5

Amount = rate × time. In litres, the first time, and the second time (d minutes longer)

6x

5(x + 3)

Practise this →

4. Formulae

Substitute into a formula: P = 2(l + b), l = 8, b = 5

\displaystyle P = 2(l + b)

e.g.

\displaystyle 2(8 + 5)

\displaystyle = 26

📖 Worked example

The formula $v = u + at$ gives the speed of a moving object. Find the value of $t$ when $v = 18$ , $u = 2$ and $a = 4$ .

Step 1 of 3

Substitute the values you know

18 = 2 + 4t

Practise this →

5. Preview (Sec 2): changing the subject

Make x the subject: undo in reverse order

\displaystyle y = 3x - 7

e.g.

\displaystyle y + 7 = 3x

\displaystyle x = \dfrac{y + 7}{3}

Make b the subject

\displaystyle P = 2(l + b)

e.g.

\displaystyle \dfrac{P}{2} = l + b

\displaystyle b = \dfrac{P}{2} - l

📖 Worked example

Make $x$ the subject of $y = -3x - 3$ .

Step 1 of 2

Get the x term on its own

y + 3 = -3x

Practise this →

Try a question: Linear equations

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