Matemáticas
What Is a Linear Equation?
Every variable to the power one, and none multiplied together. Why 3x + 2 = 11 passes the test, why xy = 6 and x² = 9 fail, and where the name comes from.
Linear equation
An equation in which every variable appears only to the power one, with no variables multiplied together, so its graph is a straight line.
- También llamado
- First-degree equation
- Dónde lo encuentra el alumnado
- Grade 7 mathematics, when equations first carry a letter on both sides, and again at GCSE and IGCSE as the equation of a straight line.
La respuesta corta
A linear equation is one in which every variable appears only to the power one and no two variables are multiplied together. 3x + 2 = 11 is linear; xy = 6 and x² = 9 are not. Plotted on axes, a linear equation in two variables always gives a straight line.
Un ejemplo
3x + 2 = 11 ✓ xy = 6 ✗ x² = 9 ✗
Only the first passes both conditions. In 3x + 2 = 11 the single variable sits at the power one, so subtracting 2 and dividing by 3 gives x = 3 and nothing else. xy = 6 keeps both variables at the power one but multiplies them together, and its graph is a curve. x² = 9 breaks the power condition outright, and it has two answers, 3 and −3, where a linear equation in one variable has one.
The test that decides it
Two conditions, and an equation has to pass both. Every variable must appear to the power one — no squares, no cubes, nothing under a root sign, nothing sitting in a denominator. And no two variables may be multiplied by one another.
Run the awkward cases through it. x² = 9 fails on the first condition and there is nothing to argue about. xy = 6 is the interesting one: both x and y are to the power one, so it passes the first condition, and it still fails, because the two variables are multiplied together.
5/x = 10 fails as well, though it looks harmless. Rewriting it as 5x⁻¹ = 10 makes the reason visible — the variable is at the power −1, not 1.
- 3x + 2 = 11 — linear
- 2x + 5y = 20 — linear, two variables
- x² = 9 — not linear, the variable is squared
- xy = 6 — not linear, two variables multiplied
- 5/x = 10 — not linear, x sits at the power −1
Why the word is about a picture
The name is borrowed from the graph rather than the algebra. Every pair of values satisfying 2x + 5y = 20 lies on one straight line; the pairs satisfying y = x² lie on a curve. That is the whole reason for the word, and it is why those two conditions are the ones that matter — break either and the graph bends.
In one variable the picture is simpler still, and it comes with a guarantee: a linear equation such as 3x + 2 = 11 has exactly one solution. There are two collapses worth recognising. 2x + 1 = 2x + 5 has no solution, because the variable vanishes and leaves a false statement, and 2x + 4 = 2(x + 2) is true for every value of x.
Linear does not mean easy
Students read the word as a comment on difficulty, which holds until it does not. 4(3x − 7)/5 + 2 = 3(x + 1) is linear — one variable, power one — and it takes several careful lines to unpick. x² = 9 is not linear and takes one.
The word classifies the equation, not the effort. What it buys you is certainty about method: any linear equation in one variable is solved by collecting terms and dividing. No factorising, no formula, and no second answer to go hunting for.
Preguntas frecuentes
Is 3x + 2y = 12 a linear equation?
Yes. Both variables are at the power one and neither multiplies the other, so it passes the test. Having two variables means it does not have one solution — it has infinitely many pairs of values, and every one of them lies on the same straight line when plotted.
Why is x² = 9 not linear?
Because the variable is squared, which breaks the power-one condition. The consequence shows up on the graph, where y = x² is a curve rather than a line, and in the answer, where x² = 9 has two solutions, 3 and −3. Linear equations in one variable have exactly one.
Is 5/x = 10 a linear equation?
No. Dividing by x is the same as multiplying by x⁻¹, so the variable sits at the power −1. Equations of that shape have curved graphs that run away towards the axes rather than straight lines, and they are usually called reciprocal or rational equations instead.
What is the difference between a linear equation and a linear expression?
An equation has an equals sign and can be solved; an expression cannot. 3x + 2 is a linear expression, and there is nothing to find — you can only simplify or substitute. 3x + 2 = 11 is a linear equation, and x = 3. Exam papers use the two words precisely.
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