Sec 1 · G2 (N(A))

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. 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

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3. 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 →

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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

Practise this →

2. 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

Practise this →

3. 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 →