Solving Linear Equations Examples

This resource goes through three examples of solving two-step equations using the balancing method and the function machine method.

Solving Linear Equations: Step-by-Step Guide with Examples

Solving Linear Equations is an important topic of the KS3 and KS4  Algebra curriculum. Your students will be able to solve and calculate unknown values by following this mathematical concept. You can either be a student who is preparing for upcoming exams or a teacher who is teaching maths, this is a perfect teaching resource for you. This ‘Solving Linear Equations: Step-by-Step Guide’ resource worksheet will help you understand the basic idea of solving linear equations in a simple, easy-to-understand way.

What is a Linear Equation?

The concept of the linear equation can be easily explained as an equation where the highest power of the variable must be 1. These equations have a straight-line graph when plotted.

\(
\textbf{Examples of Linear Equations:} \\[6pt]
8a – 5 = 11 \\[6pt]
10 + 6y = 34 \\[6pt]
\frac{x}{12} – 5 = 4
\)

There are mainly two important methods that can be used while solving linear equations. These are-

1. The Balancing Method
2. The Function Machine Method

Method 1: Solving Linear Equations Using the Balancing Method

The balancing method is a form of solving Linear Equations. The basic idea of this method involves performing the same operation on both sides of any equation to keep it balanced.

\[
\begin{aligned}
\textbf{Example 1: Solving } &\ 8a – 5 = 11 \\
\text{Step 1: Add 5 to both sides.} & \\
8a – 5 + 5 &= 11 + 5 \\
8a &= 16 \\
\\
\text{Step 2: Divide both sides by 8.} & \\
a &= \frac{16}{8} = 2 \\
\\
\textbf{Solution: } &\ a = 2
\end{aligned}
\]

Method 2: Solving Linear Equations Using the Function Machine

On the other hand, the Function Machine method is another way through which we can solve linear equations. This method works by reversing operations in the opposite order.

\[
\begin{aligned}
\textbf{Example 2: Solving } &\ \frac{x}{12} – 5 = 4 \\
\text{Step 1: Add 5 to both sides.} & \\
\frac{x}{12} – 5 + 5 &= 4 + 5 \\
\frac{x}{12} &= 9 \\
\\
\text{Step 2: Multiply both sides by 12.} & \\
x &= 9 \times 12 \\
x &= 108 \\
\\
\textbf{Solution: } &\ x = 108
\end{aligned}
\]

Key Tips for Solving Linear Equations

Here are some important tips for solving linear equations-

• We must always perform the same operation on both sides to maintain balance.
• We need to use inverse operations (addition ↔ subtraction, multiplication ↔ division).
• You must always check your answer by substituting it back into the original equation.