# Solving Linear Equations: All Types

An
**
equation
**
has to have an equal 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,
$\frac{1}{x}$
, 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

$3\left(2\right)+5=11$ , 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.

$\begin{array}{l}n+8-8=10-8\\ n=2\end{array}$

**
Example 2:
**

Solve for $y$ .

$\frac{3}{4}y=15$

The inverse operation of multiplication is division. So, divide both sides by $\frac{3}{4}$ $\left(\text{which is the same as multiplying by}\frac{4}{3}\right)$ .

$\frac{4}{3}\cdot \frac{3}{4}y=\frac{4}{3}\cdot 15$

$y=20$

## Two-Step Linear Equations

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

**
Example 3:
**

Solve for $x$ .

$3x+5=11$

$3x+5=11$ | The given equation. |

$3x+5-{5}=11-{5}$ |
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. |

$3x=6$ | We have undone one operation. One more to go. |

$\frac{3x}{{3}}=\frac{6}{{3}}$ | 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, $y=3x+2$ . This equation has not just one but infinitely many solutions; the solutions can be graphed as a line in the plane.