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Solving Linear Equations

An equation has to have an equals sign, as in 3x + 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 3x + 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.

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

n + 8 – 8 = 10 – 8

n = 2

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.

3x + 5 = 11 . . . . . . .our given equation

– 5 . . . . . . . – 5 . . . . subtract 5 from each side to get constants on the right

3x = 6 . . . . . . . . . . . the result

3x/3 = 6/3 . . . . . . . .divide both sides by 3 to isolate the x

x = 2 . . . . . . . . . . . . the solution (same as before!)

. . . . . . . . . . . . . . . . .We've solved the equation .

The thing that makes these equations linear is that the highest power of x is x¹ (no x² or other powers; for those, see quadratic equations and polynomials).

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