Linear Equations

Q: What is the solution of 3x = 18?

x = 6 — Divide both sides by 3: x = 18 ÷ 3 = 6.

Q: What is the solution of 2x + 5 = 13?

x = 4 — 2x = 13 − 5 = 8, so x = 4.

Q: What is the solution of 5x − 3 = 2x + 9?

x = 4 — 5x − 2x = 9 + 3, so 3x = 12 and x = 4.

Q: You buy 5 cakes at ¥x each and pay ¥200 for the box. The total comes to ¥2,200. What is x?

400 — 5x + 200 = 2200 → 5x = 2000 → x = 400.

Q: Fixed costs are ¥60,000 and each unit earns ¥300 of profit. How many units is the break-even point (neither profit nor loss)?

200 units — 300x = 60000, so x = 200 units.

Q: What is the solution of 4(x − 2) = 12?

x = 5 — x − 2 = 3, so x = 5. Or expand first: 4x − 8 = 12 → 4x = 20 → x = 5.

Where this is used

"How many do we have to sell before we stop losing money?" That is an equation problem. Break-even analysis, comparing two phone plans, working out speed and distance — any time you work backwards from a result, in business or in everyday life, an equation is doing the work underneath. Startup founders use equations to build their business plans.

Start gently

An equation is a balanced scale. Take the same weight off both sides and it stays balanced. That single rule is enough to find x.

Key points

What an equation is

An equation is a statement that two things are equal, contains a letter such as x, and is only true for certain values of that letter. A value of the letter that makes it true is called a solution.

Properties of equality

(1) Add the same number to both sides and the equality still holds. (2) Subtract the same number from both sides and it still holds. (3) Multiply both sides by the same number and it still holds. (4) Divide both sides by the same number (as long as it is not 0) and it still holds.

How to solve a linear equation

Move the terms containing the letter to the left side and the number-only terms to the right side, then tidy each side up. Finally, divide both sides by the coefficient.

The trick to moving terms

When you move a term to the other side, its sign flips. Remember it as "plus becomes minus, minus becomes plus."

Break-even written as an equation

A product has ¥100,000 of fixed costs and earns ¥500 of profit per unit. Sell x units, and at what x do you break even? Set up an equation and solve it.

See it drawn

Solving 3x + 4 = 13

Check yourself 10questions

Tap an option to check

Q1 What is the solution of x + 7 = 12?

Q2 What is the solution of 3x = 18?

Q3 What is the solution of 2x + 5 = 13?

Q4 What is the solution of 5x − 3 = 2x + 9?

Q5 You buy 5 cakes at ¥x each and pay ¥200 for the box. The total comes to ¥2,200. What is x?

Q6 Fixed costs are ¥60,000 and each unit earns ¥300 of profit. How many units is the break-even point (neither profit nor loss)?

Q7 What is the solution of 4(x − 2) = 12?

Q8 What is the solution of x ÷ 3 + 1 = 5?

Q9 A younger brother is 8 years younger than his older brother. Their ages add up to 40. How old is the older brother?

Q10 What is the solution of −2x + 6 = −4?

Jobs that use this

Management consultant $150k

Pharmacist $75k

Logistics planner $65k

Management consultant$150k

Analyzes how a company makes and loses money, then proposes ways to improve its profits. Break-even calculations are a basic skill of the job.

Maths used: EquationsStatisticsGraphsOptimization

Pharmacist$75k

Calculates drug doses to match a patient's body weight and kidney function. Working backwards with equations is part of the daily routine.

Maths used: EquationsProportionConcentration calculationsStatistics

Logistics planner$65k

Uses speed, distance, and time calculations to optimize delivery routes and how long they take.

Maths used: EquationsSpeed, time, and distanceOptimizationGraph theory