Definition
Linear equation
A linear equation has variables only to the first power, with no variables multiplied together.
Continue from this idea into connected AXIO math.
Defining relationship
First power, no variable-times-variable; two-variable solutions line up straight.
A linear equation can be written as a sum of constant multiples of variables equal to a constant, with at least one variable coefficient nonzero. In one variable it has the form ax + b = c; in two variables its solution set graphs as a straight line.
First powers produce a straight solution line here.
What it means
Linear means the variable parts do not bend into squares, cubes, or products such as xy. In one variable, solving keeps the same value on both sides while you undo operations. In two variables, all solution points lie on a straight line.
Remember it
First power, no variable-times-variable; two-variable solutions line up straight.
Example
Solve 4x − 7 = 13. Add 7 to both sides: 4x = 20. Divide both sides by 4: x = 5. Checking gives 4 × 5 − 7 = 13, so 5 is the solution.
Watch for
x² + 1 = 10 is not linear because x is squared; xy = 6 is not linear because variables are multiplied.
Common mix-up
A linear equation can include brackets or fractions; what matters is whether it can be rewritten in linear form.
Exact meaning
A linear equation can be written as a sum of constant multiples of variables equal to a constant, with at least one variable coefficient nonzero. In one variable it has the form ax + b = c; in two variables its solution set graphs as a straight line.