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2. Linear Relations and Equations
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VCE 11 General 2023

2.03 Solve linear equations

Lesson
Worksheet
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Lesson

Solve linear equations

The aim of solving a linear equation is to work out the unknown value, such as x. To solve for the unknown, we want to transpose the equation so that x is the subject of the equation. To do this, we apply inverse (opposite) operations to both sides.

The most important thing to remember is that both sides of the equation must remain balanced. Always remember; any operation applied to one side of an equation must be applied identically to the other side of the equation.

To double check if an answer is correct, we can substitute the answer back into the original equation. If the two sides of the equation are equal, then the answer found must be correct.

Examples

Example 1

Solve the following equation: 8m+9=65.

Worked Solution
Create a strategy

Use inverse operations to make m the subject.

Apply the idea
\displaystyle 8m+9-9\displaystyle =\displaystyle 65-9Subtract 9 from both sides
\displaystyle 8m\displaystyle =\displaystyle 56Evaluate the subtraction
\displaystyle \dfrac{8m}{8}\displaystyle =\displaystyle \dfrac{56}{8}Divide both sides by 8
\displaystyle m\displaystyle =\displaystyle 7Evaluate

Example 2

Solve \dfrac{x+9}{7}=4

Worked Solution
Create a strategy

Use inverse operations to make x the subject.

Apply the idea

The first thing we should do is multiply both sides by 7 to cancel out the denominator.

\displaystyle \dfrac{x+9}{7} \times 7\displaystyle =\displaystyle 4 \times 7Multiply both sides by 7
\displaystyle x+9\displaystyle =\displaystyle 28Simplify
\displaystyle x+9-9\displaystyle =\displaystyle 28-9Subtract 9 from both sides
\displaystyle x\displaystyle =\displaystyle 19Evaluate the subtraction

Example 3

Solve 4\left(5x+1\right)=-3\left(5x-5\right)

Worked Solution
Create a strategy

Expand the brackets then simplify like terms.

Apply the idea
\displaystyle 20x+4\displaystyle =\displaystyle -15x+15Expand both brackets
\displaystyle 20x+4+15x\displaystyle =\displaystyle -15x+15+15xAdd 15x on both sides
\displaystyle 35x+4\displaystyle =\displaystyle 15Combine like terms
\displaystyle 35x+4-4\displaystyle =\displaystyle 15-4Subtract 4 from both sides
\displaystyle 35x\displaystyle =\displaystyle 11Combine like terms
\displaystyle \dfrac{35x}{35}\displaystyle =\displaystyle \dfrac{11}{35}Divide both sides by 35
\displaystyle x\displaystyle =\displaystyle \dfrac{11}{35}Evaluate
Idea summary

The aim of solving a linear equation is to work out the unknown value, such as x. To solve for the unknown, we want to transpose the equation so that x is the subject of the equation. To do this, we apply inverse (opposite) operations to both sides.

Solve linear equations using technology

Linear equations can also be solved using technology, such as a CAS calculator. The video below explains how to use a CAS calculator to quickly produce a solution to any given linear equation.

Examples

Example 4

Solve the following linear equation using technology.\frac{5x-15}{4}=-20

Worked Solution
Create a strategy

Using technology such as calculator, enter the equation and solve for x.

Apply the idea

x=-13

Idea summary

Linear equations can also be solved using technology, such as a CAS calculator.

Outcomes

U1.AoS4.1

the properties of linear functions and their graphs

U1.AoS4.8

solve linear equations constructed from word problems, including simultaneous linear equations

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