By Thread Academy · 22 September 2026 · Mathematics
Equations are the sentences of algebra, and linear equations are the simplest and most useful of them all. Every time you work out the cost of a phone plan, convert a temperature, or predict how far you can travel on a tank of fuel, a linear equation is quietly doing the work in the background.
This guide explains exactly what a linear equation is, why mathematicians call it "linear", the standard forms you will meet at IGCSE, and how to recognise and interpret them. By the end, you will see straight lines everywhere.
What makes an equation linear
An equation is a mathematical statement that two expressions have the same value, joined by an equals sign. An equation is linear when every variable in it appears to the power of one only. That single rule is the whole definition.
A linear equation must never contain:
- a variable raised to a power other than one, such as x squared or x cubed
- the product of two variables multiplied together, such as xy
- a variable inside a root, such as the square root of x
- a variable sitting in a denominator, such as 3 divided by x
So these are all linear:
3x + 5 = 20 y = 2x - 7 4a - b = 11
And these are not:
x^2 + 3x = 10 (x appears squared) y = 1/x + 4 (x sits in a denominator) p + q = pq (the term pq multiplies two variables)
Worked example 1: spot the linear equations
Which of these equations are linear in their variables?
a) 5t - 8 = 2t + 7 b) m^2 = 4m + 12 c) 2x + 3y = 6 d) z = sqrt(z + 1) + 2
Step 1: Check the power of each variable. In (a), t appears only to the first power, so it is linear. In (b), m is squared, so it is not linear.
Step 2: Look for products and roots. In (c), x and y each appear to the first power and are never multiplied together, so it is linear. In (d), z appears inside a square root, so it is not linear.
Answer: (a) and (c) are linear.
Why is it called "linear"?
The word "linear" comes from "line", and it earns its name on a graph. If you take a linear equation in two variables, find pairs of values that satisfy it, and plot them as coordinates, every point falls on a single straight line.
Take y = 2x + 1. Substituting x values gives a table of solutions:
x = -1 gives y = -1 x = 0 gives y = 1 x = 1 gives y = 3 x = 2 gives y = 5
Plotting the points (-1, -1), (0, 1), (1, 3) and (2, 5) produces points that lie perfectly on a straight line. That always happens with linear equations, and it never happens with non-linear ones. The equation y = x^2, for example, gives points that fall on a curve. So the name is a promise: linear equations draw straight lines.
The standard forms you will meet
Linear equations appear in several standard disguises. Recognising each form helps you read off useful information quickly.
Slope-intercept form: y = mx + c. The number m is the gradient, which tells you how steep the line is, and c is the y-intercept, the point where the line crosses the y-axis. For example, in y = 3x - 2 the gradient is 3 and the line crosses the y-axis at -2.
General form: ax + by + c = 0, where a and b are not both zero. This is the most flexible form because it can describe every straight line, including vertical ones. For example, 2x - 5y + 10 = 0.
Special cases. The equation y = 4 is a horizontal line passing through 4 on the y-axis, and x = -3 is a vertical line passing through -3 on the x-axis. Both are linear because the highest power of any variable is one.
Worked example 2: converting between forms
Rewrite 4x + 2y = 12 in slope-intercept form, and state the gradient and y-intercept.
Step 1: Isolate the y term by subtracting 4x from both sides:
2y = -4x + 12
Step 2: Divide every term by 2:
y = -2x + 6
Step 3: Compare with y = mx + c. The gradient m is -2 and the y-intercept c is 6. The negative gradient tells us the line slopes downwards from left to right.
Linear equations in real life
Many everyday situations have a fixed starting amount plus a steady rate of change, and those are exactly linear.
A taxi fare. A taxi charges a 3 dollar flag-fall plus 2 dollars per kilometre. If d is the distance travelled, the fare F is:
F = 2d + 3
This is linear in d. The constant 3 is the starting charge and the coefficient 2 is the rate per kilometre.
A phone plan. A monthly plan costs 10 dollars plus 5 dollars for each gigabyte of data used. If g is the gigabytes used, the monthly cost C is:
C = 5g + 10
Worked example 3: interpreting the numbers
A catering company quotes P = 8n + 150, where P is the total price in dollars and n is the number of guests. Explain what the numbers 8 and 150 represent.
Step 1: Compare with y = mx + c. The coefficient of n is 8, which plays the role of the gradient, the rate of change.
Step 2: Interpret the coefficient. Each extra guest adds 8 dollars, so 8 is the price per guest.
Step 3: Interpret the constant. When n is 0, P is 150, so 150 is a fixed booking fee charged regardless of guest numbers.
Key takeaways
- A linear equation is one where every variable appears to the power of one only, with no powers, roots, products of variables, or variables in denominators.
- It is called linear because its solutions always form a straight line when graphed.
- Know the standard forms: slope-intercept form y = mx + c reveals the gradient and intercept, while general form ax + by + c = 0 can describe any line.
- Real-life situations with a fixed amount plus a constant rate are modelled by linear equations.
- To interpret a linear model, match it against y = mx + c: the coefficient of the variable is the rate per unit, and the constant is the starting value.