Solving Linear Equations

Solving linear equations is a fundamental skill in Maths. It involves finding the value of the unknown variable that satisfies the equation. In this guide, we will cover key topics including equations and identities, number machines, solving equations, solving equations with brackets, solving equations with unknowns on both sides, and solving equations with fractions. By mastering these concepts, you will be able to solve a variety of linear equations confidently.

Equations and Identities

Equations

An equation is a mathematical statement that shows the equality of two expressions, usually involving an unknown variable. The goal is to solve for the unknown variable.

For example:

• $x + 5 = 12$, where the unknown is $x$.

Identities

An identity is an equation that is always true, no matter what values are substituted for the variable. Unlike an equation, which has a specific solution, an identity holds true for all values.

For example:

• $2(x + 3) = 2x + 6$ is an identity because both sides are always equal for any value of $x$.

Key Difference:

• Equation: Has a solution (e.g. $x + 5 = 12$).
• Identity: True for all values (e.g. $2(x+3)=2x+6$).

Number Machines

A number machine can be a helpful way of thinking about solving equations. Imagine a machine that performs certain operations on a number (like adding, subtracting, multiplying, or dividing), and then outputs a result. You can work backwards to find the input (the unknown).

For example, imagine a number machine that:

1. Multiplies a number by 3.
2. Adds 7 to the result.
3. Outputs 22.

This can be written as the equation:

$$3x + 7 = 22$$

Your task is to find the number $x$ that, when passed through the machine, gives the output of 22. By solving the equation, you find that $x = 5$.

Solving Simple Equations

To solve a simple linear equation, the goal is to isolate the unknown variable (usually xxx) on one side of the equation.

Example 1: Solving a basic equation

Solve $x + 6 = 10$.

• Step 1: Subtract 6 from both sides of the equation to isolate $x$: 

  $$x = 10 – 6$$

• Step 2: Simplify: 

  $$x = 4$$

So, the solution is $x = 4$.

Solving Equations with Brackets

When solving equations with brackets, you need to expand the brackets first, then simplify the equation before solving for the unknown.

Example 2: Solving an equation with brackets

Solve $2(x + 4) = 18$.

• Step 1: Expand the brackets by multiplying 2 by both $x$ and 4: 

  $$2x + 8 = 18$$

• Step 2: Subtract 8 from both sides to isolate the term with $x$: 

$$2x = 18 – 8$$

$$2x = 10$$

• Step 3: Divide both sides by 2 to solve for $x$:

$$x = \frac{10}{2}$$

$$x=5$$

So, the solution is $x = 5$.

Solving Equations with Unknowns on Both Sides

When you have unknowns on both sides of the equation, you need to collect all the terms involving the unknown variable on one side and constants on the other side.

Example 3: Solving an equation with unknowns on both sides

Solve $3x + 4 = 2x + 9$.

• Step 1: Subtract $2x$ from both sides to get the terms involving xxx on one side: 

$$3x - 2x + 4 = 9$$

$$x + 4 = 9$$

• Step 2: Subtract 4 from both sides to isolate $x$: 

$$x=9−4$$

$$x = 5$$

So, the solution is $x = 5$.

Solving Equations with Fractions

Solving equations with fractions may seem tricky, but the key is to eliminate the fractions by multiplying both sides of the equation by the denominator.

Example 4: Solving an equation with fractions

Solve $\frac{x}{3} + 2 = 5$.

• Step 1: Subtract 2 from both sides to isolate the fraction: 

$$\frac{x}{3} = 5 – 2$$

$$\frac{x}{3} = 3$$

• Step 2: Multiply both sides by 3 (the denominator) to eliminate the fraction: 

$$x = 3 \times 3$$

$$x=9$$

So, the solution is $x = 9$.

Example 5: Solving an equation with more fractions

Solve $\frac{2x}{5} - 4 = 6$.

• Step 1: Add 4 to both sides to isolate the term with $x$: 

$$\frac{2x}{5} = 6 + 4$$

$$\frac{2x}{5} = 10$$

• Step 2: Multiply both sides by 5 to eliminate the fraction: 

$$2x = 10 \times 5$$

$$2x=50$$

• Step 3: Divide both sides by 2 to solve for $x$: 

$$x = \frac{50}{2}$$

$$x=25$$

So, the solution is $x = 25$.

Summary of Key Steps

• Solving basic equations: Isolate the variable using inverse operations (addition/subtraction, multiplication/division).
• Solving equations with brackets: Expand the brackets, then solve.
• Solving equations with unknowns on both sides: Move all the unknown terms to one side and constants to the other.
• Solving equations with fractions: Multiply both sides by the denominator to eliminate fractions, then solve.

By following these steps and practising regularly, you’ll be able to confidently solve a variety of linear equations, an essential skill for your GCSE Maths exam.