Linear equations and real-life problems – Equations and Inequalities (Senior 4 Basic Mathematics) | Ganzaa

Equations and Inequalities

Lessons

Senior 4 · Basic Mathematics

Equations and Inequalities

5 lessons

Lesson 1 of 5

Linear equations and real-life problems

25 min

How far did the passenger ride?

Suppose a motorcycle-taxi fare is worked out as a fixed charge of 300 Frw plus 150 Frw for every kilometre (illustrative rates). A passenger pays 1 950 Frw. How far did she ride? Call the distance $d$ kilometres. The fare is $300 + 150 d$ , so

300 + 150 d = 1950.

Taking 300 from both sides gives $150 d = 1650$ , and dividing by 150 gives $d = 11$ . She rode 11 km. Almost every word problem in business, science or daily life becomes an equation like this one. This lesson gives you a reliable method for solving linear equations, and for building them from words.

Definition

Equation: a statement that two expressions are equal. A linear equation in one unknown $x$ can be written as

ax + b = 0, \qquad a \ne 0,

and has exactly one solution (root), $x = -\frac{b}{a}$ . The solution set is the set of all solutions, written $S = \left\{ -\frac{b}{a} \right\}$ .

Equivalent equations have the same solutions. You get one by adding the same number to both sides, or by multiplying or dividing both sides by the same non-zero number.

Example: $4 x - 10 = 0$ gives $x = \frac{10}{4} = \frac{5}{2}$ .

A method for any linear equation

Clear fractions: multiply every term by the LCM of the denominators.
Remove brackets, taking care with a minus sign in front of a bracket.
Collect the $x$ -terms on one side and the numbers on the other.
Divide by the coefficient of $x$ .
Check by substituting into the original equation.

Worked example 1: brackets and fractions

Solve (a) $3 (2 x - 5) - 4 (x + 1) = 7$ (b) $\frac{x + 2}{3} - \frac{x - 1}{4} = 3$ .

Solution

$6 x - 15 - 4 x - 4 = 7$ , so $2 x - 19 = 7$ , $2 x = 26$ , $x = 13$ . Check: $3 (21) - 4 (14) = 63 - 56 = 7$ . ✓
Multiply every term by 12: $4 (x + 2) - 3 (x - 1) = 36$ , so $4 x + 8 - 3 x + 3 = 36$ , $x + 11 = 36$ , $x = 25$ . Check: $\frac{27}{3} - \frac{24}{4} = 9 - 6 = 3$ . ✓

Warning

The minus sign in front of a bracket changes every sign inside it. $- 3 (x - 1) = - 3 x + 3$ , not $- 3 x - 3$ . And when you clear fractions, multiply every term, including the whole numbers: in (b) above, the 3 on the right becomes 36.

Important

Not every equation has exactly one solution. If the $x$ -terms cancel, look at what is left.

$2 (x + 3) = 2 x + 5$ gives $6 = 5$ : false, so there is no solution, $S = \varnothing$ .
$2 (x + 3) = 2 x + 6$ gives $6 = 6$ : true for every $x$ , so every real number is a solution, $S = \mathbb{R}$ .

From words to an equation

To solve a problem in words: (1) say what the unknown stands for, with its unit; (2) write each quantity in terms of it; (3) write the equation; (4) solve; (5) check the answer in the story, not just in the equation; (6) answer in a sentence.

Worked example 2: break-even

A juice maker has fixed costs of 60 000 Frw a month (rent and a blender). Each bottle costs 400 Frw to make and sells for 700 Frw. How many bottles must she sell in a month to break even, that is, for income to equal costs?

Solution

Let $n$ be the number of bottles. Income $= 700 n$ ; costs $= 60\,000 + 400n$ .

700n = 60\,000 + 400n \;\Longrightarrow\; 300n = 60\,000 \;\Longrightarrow\; n = 200 .

Check: income $700 \times 200 = 140\,000$ ; costs $60\,000 + 80\,000 = 140\,000$ . ✓ She breaks even at 200 bottles; every bottle above that earns 300 Frw of profit.

Seeing the break-even point

Each side of the equation $700n = 60\,000 + 400n$ is a straight line. The solution of the equation is the $n$ -coordinate of the point where the lines cross. Below 200 bottles the cost line is above the income line (a loss); above 200 the income line is higher (a profit).

Worked example 3: sharing and mixing

(a) Three sisters share 120 000 Frw. The second gets 10 000 Frw more than the first, and the third gets twice as much as the first. How much does each get? (b) How much water must be added to 2 litres of a juice concentrate that is 40% fruit to make a drink that is 25% fruit?

Solution

Let the first sister get $x$ Frw. Then $x + (x + 10\,000) + 2x = 120\,000$ , so $4x = 110\,000$ and $x = 27\,500$ . The shares are 27 500, 37 500 and 55 000 Frw (total 120 000 ✓).
The fruit content does not change: $0.40 \times 2 = 0.8$ L. With $w$ litres of water added, the drink is $2 + w$ litres, so $0.25 (2 + w) = 0.8$ , $0.5 + 0.25 w = 0.8$ , $w = 1.2$ . Add 1.2 litres of water.

Worked example 4: changing the subject of a formula

The amount of a simple-interest investment is $A = P (1 + r t)$ . Make $t$ the subject, and find how long 400 000 Frw takes to grow to 496 000 Frw at 6% a year.

Solution

$A = P + P r t$ , so $P r t = A - P$ and

t = \frac{A - P}{Pr} = \frac{496\,000 - 400\,000}{400\,000 \times 0.06} = \frac{96\,000}{24\,000} = 4 \text{ years}.

Tip

Write the unknown's meaning and unit at the top of your working: "Let $n$ = number of bottles sold per month". It stops you mixing up quantities, and it tells you how to write the final sentence. A negative number of bottles, or a share bigger than the total, means an error somewhere.

3minutemaths: How to solve linear equations with fractions and brackets. Watch how multiplying every term by the LCM clears the fractions in one step.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Solve (a) $5 x - 7 = 3 x + 9$ (b) $2 (3 - x) = 4 - (x - 5)$ .
Solve $\frac{2x - 1}{5} = \frac{x + 4}{3}$ .
Solve $\frac{x}{2} + \frac{x}{3} - \frac{x}{4} = 14$ .
Say whether each equation has one solution, none, or infinitely many: (a) $3 (x - 2) = 3 x - 6$ (b) $4 x + 1 = 4 (x + 1)$ .
A phone shop pays a worker 40 000 Frw a month plus 2 500 Frw for every phone she sells. One month she earned 102 500 Frw. How many phones did she sell?
The sum of three consecutive even numbers is 162. Find them.
A cooperative's fixed costs are 150 000 Frw a month. It makes baskets for 1 800 Frw each and sells them for 4 300 Frw. How many baskets must it sell to make a profit of 100 000 Frw?

Show answersHide answers

(a) $2 x = 16$ , $x = 8$ (b) $6 - 2 x = 9 - x$ , so $x = - 3$ .
$3 (2 x - 1) = 5 (x + 4)$ , $6 x - 3 = 5 x + 20$ , $x = 23$ .
Multiply by 12: $6 x + 4 x - 3 x = 168$ , $7 x = 168$ , $x = 24$ .
(a) $- 6 = - 6$ : infinitely many (every real number). (b) $1 = 4$ : no solution.
$40\,000 + 2\,500p = 102\,500$ , so $p = 25$ phones.
$n + (n + 2) + (n + 4) = 162$ , $3 n = 156$ , $n = 52$ : 52, 54 and 56.
$4300b - (150\,000 + 1800b) = 100\,000$ , so $2500b = 250\,000$ and $b = 100$ baskets.

Quick check

Key points

$a x + b = 0$ with $a \ne 0$ has the single solution $x = -\frac{b}{a}$ .
Clear fractions, expand, collect, divide, check.
If $x$ cancels: a false statement means no solution; a true one means every real number.
In word problems, define the unknown with its unit and check the answer in the story.

Sources and credits

Written by Ganzaa: the explanations, the break-even graph, the worked examples, the practice items and the questions. The unit follows the topic list of REB's S4 Basic Mathematics (2026–2027); no REB text is reproduced. All prices and fares are illustrative.
Photo: A Moto (motorcycle taxi) in a street in Kigali, Jenny Paul, Wikimedia Commons, CC BY-SA 4.0.
Video: How to solve linear equations with fractions and brackets, 3minutemaths, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1How far did the passenger ride?

2A method for any linear equation

3From words to an equation

4Seeing the break-even point

How far did the passenger ride?

A method for any linear equation

From words to an equation

Seeing the break-even point