Linear Equations

An equation that can be written in the form

a x + b = 0, (a \neq 0)

where $a$ and $b$ are (fixed) real numbers and $x$ is the variable (or the unknown) is called a linear equation.

Quick Reference

Form	Solution	Notes
$a x + b = 0$	$x = - \frac{b}{a}$	$a \neq 0$
With fractions	Multiply both sides by the LCD, then solve	Clears all denominators at once

Solving a Linear Equation

To solve the equation $a x + b = 0$ , subtract $b$ from both sides

ax+\cancel{b}-\cancel{b} = -b

and then divide both sides of the resulting equation, $a x = - b$ , by $a$ :

x = - \frac{b}{a}

(Sometimes, this process is described as transferring $b$ to the other side of the equation and then moving the coefficient $a$ to the denominator of $b$ .)

Examples

Example 1. Solve: $5 x + 9 = 8 x + 12$ .

Solution

We first simplify and then rewrite the equation so that all terms containing the variable

x

are on one side and the constant terms are on the other side. \begin{aligned} 5x+9-8x & =8x-8x+12 &&(\text{subtract } 8x \text{ from both sides})\\ -3x+9 & =\cancel{0x}+12\\ -3x+9-9 & =12-9 &&(\text{subtract } 9 \text{ from both sides})\\ -3x & =3\\ x & =-1&&(\text{divide both sides by } {-3}) \end{aligned}

Example 2. Solve $\frac{2 x}{3} - \frac{x - 2}{2} = \frac{x}{6} - (4 - x)$ .

Solution

To clear the fractions, multiply both members by the LCD, 6:

4 x - 3 (x - 2) = x - 6 (4 - x)

4 x - 3 x + 6 = x - 24 + 6 x

Transpose and collect terms:

- 6 x = - 30

. Therefore

x = 5.

Verification:

\frac{2 \cdot 5}{3} - \frac{5 - 2}{2} \equiv \frac{5}{6} - (4 - 5) .

Example 3. Solve $\frac{3 x + 7}{4} = 1 + \frac{1 - x}{3}$ .

Solution

\begin{aligned} \dfrac{3x+7}{4} & =1+\dfrac{1-x}{3}&&\text{(given equation)}\\ 12\cdot\frac{3x+7}{4} & =12\cdot\left(1+\frac{1-x}{3}\right)&&\text{(multiply both sides by $LCD = 12$)}\\ 3\cdot(3x+7) & =12+4\cdot(1-x)&&\text{(simplify)}\\ 9x+21 & =12+4-4x&&\text{(expand)}\\ 9x+4x & =12+4-21&&\text{(add $4x$ to and subtract $21$ from both sides)}\\ 13x & =-5&&\text{(simplify)}\\ x & =-\frac{5}{13}&&\text{(divide both sides by $13$)} \end{aligned}

Frequently Asked Questions

What is a linear equation?

A linear equation is an equation that can be written in the form

a x + b = 0

, where

a \neq 0

and

a

b

are real numbers. It is called "linear" because its graph,

y = a x + b

, is a straight line. A linear equation in one unknown always has exactly one solution:

x = - \frac{b}{a}

How do you solve a linear equation?

Collect all terms containing the unknown

x

on one side of the equation and all constant terms on the other side. Then divide by the coefficient of

x

. For example, to solve

a x + b = 0

: subtract

b

from both sides to get

a x = - b

, then divide by

a

to get

x = - \frac{b}{a}

How do you solve a linear equation that contains fractions?

Identify the least common denominator (LCD) of all the fractions in the equation. Multiply every term on both sides by the LCD to clear all denominators at once. This produces a simpler equation with no fractions, which you can solve by the standard method.

What is the geometric meaning of the solution?

The solution

x = - \frac{b}{a}

of the equation

a x + b = 0

is the $x$ -intercept of the straight line

y = a x + b

, that is, the point where the line crosses the

x

-axis.

How do I verify my solution?

Substitute the value you found back into the original equation and check that both sides are equal. If they are equal, the solution is correct. Always verify using the original equation, not an intermediate step, to catch any errors introduced during solving.

Linear Equations

Linear Equations

Quick Reference

Solving a Linear Equation

Examples

Frequently Asked Questions

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