How to solve Linear Equations In Algebra

This post explains how to solve linear equations. It contains plenty of examples and practice problems.

Linear equations definitions

A linear equation is an equation in which the highest power of the variable is 1.

For example, 2x + 5 = 8, 5x = 2x + 1 and y/2 +1 = 0 are linear equations 

whereas x² + 2x + 1= 0, x³ = 8, 1/x =√x are not linear equations.

Sides of an equation

The left-hand side (LHS) of an equation is on the left of the = sign.

The right-hand side (RHS) of an equation is on the right of the = sign.

The solution of an equation

The solution of a linear equation is the value of the variable which makes the equation true.

In other words, it makes the left-hand side equal to the right-hand side.

How to Solve Linear Equations

First of all, you should know the rules for solving equations:

1 – If you add or subtract the same algebraic expression to or from both sides of the equation, the resulting equation is equivalent to the original equation.

Example: 

a)    x − 3 = 5

x − 3 + 3 = 5 + 3    <<< adding 3 to both sides

          x   =   8        <<<simplify

Check your solution: We put x = 8 in x − 3 = 5:

8 − 3 = 5 is true. ✅✅✅

b)   x + 7 = 12

x + 7 − 7 = 12 − 7  <<<subtract 7 from both sides

         x    =   5        <<<simplify

Check your solution: We put x = 5 in x + 7 = 12:

5 + 7 = 12 is true. ✅✅✅

2 – If you multiply or divide both sides of the equations by the same non-zero algebraic expression, the resulting equation is equivalent to the original equation.

Example: Solve for x:

a)

Check your solution:

15/3 = 5 is true. ✅✅✅

b)

Check your solution:

2 × 6 = 12 is true. ✅✅✅

c) 5x + 8 = 19

   ∴  5x + 8 − 8 = 19 − 8    <<< subtracting 8 from both sides

    ∴           5x   =  11      <<< simplify

      ∴         x     =  11/5   <<< dividing both sides by 5

Check your solution:

5 × 11/5 + 8 = 11 + 8 = 19 is true. ✅✅✅

d) 

Check your solution: ✅✅✅

Note: Always check that the answer makes the original linear equation true.

3 – If an equation contains brackets, remove all brackets by applying the distributive law of multiplication over addition or subtraction.

Example 1: Sovle for x:   5(x + 1) − 2x = −7

5(x + 1) − 2x = −7

We expand brackets >>> 5x + 5 − 2x = −7

Then we collect like terms >>>  3x + 5 = −7

We subtract 5 from both sides >>> 3x + 5 − 5 = −7 − 5

Then we simplify >>>                    3x = − 12

We divide both sides by 3 >>>        x = − 12/3

Therefore,   x = − 4.

✅✅✅

Example 2: Solve for x: 

6(1 − 2x) = − 4 − 7x

We expand brackets >>>  6 − 12x =  − 4 − 7x

Then we add 7x to both sides >>>  6 − 12x + 7x = − 4 − 7x + 7x

We simplify >>>                              6 − 5x = − 4

We subtract 6 from both sides >>> 6 − 6 − 5x  =  − 4 − 6

We simplify >>>                            − 5x           = − 10

Then we divide both sides by − 5 >>>    x   =  − 10 / (− 5) 

Therefore, x = 2.

✅✅✅

Example 3: Solve for x: 

 3(x − 2) = 9 − 4(2x + 1)

We remove all brackets >>>  3x − 6 = 9 − 8x − 4

                                        ∴    3x + 8x  =  5 + 6

                                        ∴        11x    =  11

                                          ∴         x    =   1

✅✅✅

Solving Fractional Equations

To solve fractional equations, eliminate the denominators by multiplying both sides by the Lowest Common Multiple (LCM) of all the denominators. 

Then solve the resulting linear equation.

Example 1: Solve for x:

✅✅✅

Example 2: Solve for x:

✅✅✅

Example 3: Solve for x:

✅✅✅

Summary: Solving linear equations

In fact, solving linear equations is just like solving for x = something.

Here are some things we can do to solve linear equations:

• Add or subtract the same value from both sides
• Clear out any fractions by multiplying every term by the denominator
• Divide every term by the same nonzero value
• Combine like terms
• Expand brackets

Note: Always check that the answer makes the original linear equation true.

How to check your solutions

Take the solutions and put them in the original equation to see if the equation is true.

Read more 

Solving equations (basic)

How to solve quadratic equations – 3 methods – Step by step

Quadratic factorisation – 7 methods – Step by step