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Solving Linear Equations: All Types

An equation has to have an equal sign, as in 3 x + 5 = 11 .

A linear equation is one where the variable(s) are multiplied by numbers or added to numbers, with nothing more complicated than that (no exponents, square roots, 1 x   , or any other funny business).

A solution to an equation is a number that can be plugged in for the variable to make a true number statement.

For example, substituting 2 for x in 3 x + 5 = 11 gives

3 ( 2 ) + 5 = 11 , which says 6 + 5 = 11 ; that's true! So 2 is a solution.

But how do we start with the equation, and get (not guess) the solution?

One-Step Linear Equations

Some linear equations can be solved with a single operation. For this type of equation, use the inverse operation to solve.

Example 1:

Solve for n .

n + 8 = 10

The inverse operation of addition is subtraction. So, subtract 8 from both sides.

n + 8 − 8 = 10 − 8 n = 2

Example 2:

Solve for y .

3 4 y = 15

The inverse operation of multiplication is division. So, divide both sides by 3 4 ( which is the same as multiplying by 4 3 ) .

4 3 ⋅ 3 4 y = 4 3 ⋅ 15

y = 20

Two-Step Linear Equations

More commonly, we need two operations to solve a linear equation.

Example 3:

Solve for x .

3 x + 5 = 11

3 x + 5 = 11 The given equation.
3 x + 5 − 5 = 11 − 5

To isolate the variable, we follow the order of operations in reverse. We undo the addition before we undo the multiplication.

Subtract 5 from both sides.
3 x = 6 We have undone one operation. One more to go.
3 x 3 = 6 3 Divide both sides by 3 .
x = 2 We have solved the equation!

 

The thing that makes these equations linear is that the highest power of x is x 1 (no x 2 or other powers; for those, see quadratic equations and polynomials ).

Other linear equations have more than one variable: for example, y = 3 x + 2 . This equation has not just one but infinitely many solutions; the solutions can be graphed as a line in the plane.