Linear equations and formulas | Year 11

01 / LEARN

Understand the idea.

Keep the equation balanced

An equation makes a claim of equality. Keep that claim true by applying the same operation to both sides. To isolate a variable, undo addition or subtraction before undoing the multiplication around it.

A solution must satisfy the original equation. Substitution checks both your arithmetic and your rearrangement: calculate the original left side using your proposed value and compare it with the right side.

3 x + 7 = 25

Choose the subject of a formula

In a practical formula, identify the unknown and attach units to the known quantities before calculating. In $C = F + r d$ , the fixed fee $F$ is paid once; the term $r d$ depends on distance.

To recover distance, remove the fixed fee before dividing by the rate: $d = \frac{C - F}{r}$ . This requires $r \neq 0$ . The numerator is the distance-dependent charge, so dividing dollars by dollars per kilometre produces kilometres.

\begin{array}{c} C = F + r d \\ ⟹ d = \frac{C - F}{r}, r \neq 0 \end{array}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Solve and check

Solve $3 x + 7 = 25$ .

Worked solution

1
Remove the added seven.
$3 x = 25 - 7 = 18$
Subtracting seven from both sides preserves equality.
2
Undo multiplication by three.
$x = \frac{18}{3} = 6$
Every term on both sides is divided by the same nonzero number.
3
Check in the original equation.
$3 (6) + 7 = 18 + 7 = 25$
The proposed value makes the original sides equal.

Answer

$x = 6$

EXAMPLE 02

Reverse a delivery rule

A delivery costs $C = 4 + 2.5 d$ dollars for a distance of $d$ km. A bill is 24 dollars. Find the distance.

Worked solution

1
Substitute the bill amount.
$24 = 4 + 2.5 d$
The formula gives the total charge, including the fixed fee.
2
Remove the fixed fee.
$2.5 d = 24 - 4 = 20$
Only the remaining twenty dollars pays for distance.
3
Divide by the distance rate.
$d = \frac{20}{2.5} = 8 km$
The rate is 2.50 dollars for each kilometre.
4
Verify the delivery charge.
$4 + 2.5 (8) = 24$
Eight kilometres reproduces the given bill.

Answer

8 km

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $5 x - 4 = 31$ .

Show answer & reasoning

$x = 7$ .

Add four to both sides to obtain $5 x = 35$ , then divide by five: $x = 7$ . Substitution gives $5 (7) - 4 = 31$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Solve $4 x + 9 = 37$ .
2 marks
Solve $7 - 2 x = - 5$ .
2 marks
Solve $3 (x + 2) = 21$ and check.
2 marks
Make $h$ the subject of $A = \frac{b h}{2}$ , where $b \neq 0$ .
2 marks
Use $C = 18 + 6 n$ to find the cost for five items.
2 marks
A tank has $V = 150 - 12 t$ litres after $t$ minutes. When does it hold 90 L?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $4 x + 9 = 37$ .
2 marks
Subtract 9: $4 x = 28$ .
Divide by 4.
$x = 7$ .
Solve $7 - 2 x = - 5$ .
2 marks
Subtract 7: $- 2 x = - 12$ .
Divide by $- 2$ .
$x = 6$ .
Solve $3 (x + 2) = 21$ and check.
2 marks
Divide by 3: $x + 2 = 7$ .
Subtract 2; $3 (5 + 2) = 21$ .
$x = 5$ .
Make $h$ the subject of $A = \frac{b h}{2}$ , where $b \neq 0$ .
2 marks
Multiply both sides by 2.
Divide the entire expression $2 A$ by $b$ .
$h = \frac{2 A}{b}$ .
Use $C = 18 + 6 n$ to find the cost for five items.
2 marks
$C = 18 + 6 (5) = 48$ .
48 dollars.
A tank has $V = 150 - 12 t$ litres after $t$ minutes. When does it hold 90 L?
2 marks
$90 = 150 - 12 t$ gives $12 t = 60$ .
Thus $t = 5$ .
5 min.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

In this lesson

Understand the idea.

Keep the equation balanced

Choose the subject of a formula

See the method in action.

Solve and check

Remove the added seven.

Undo multiplication by three.

Check in the original equation.

Reverse a delivery rule

Substitute the bill amount.

Remove the fixed fee.

Divide by the distance rate.

Verify the delivery charge.

Try it yourself.

Check your understanding

Keep practising