Linear equations 2 Revision Notes for AQA GCSE Maths

Linear equations 2

Linear equations 2 builds on basic equation solving by introducing more complex situations. You'll learn to handle equations with brackets and equations where the unknown letter appears on both sides of the equals sign.

Equations with brackets

When you encounter equations containing brackets, always start by expanding (multiplying out) the brackets before attempting to solve the equation.

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Key process:

Expand the brackets first using the distributive property
Collect like terms
Solve using inverse operations
Work through one operation at a time

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Worked Example: Solving with Brackets

Solve $2(3y + 5) = 22$

Step 1: Expand the brackets $2(3y + 5) = 6y + 10 = 22$

Step 2: Solve the resulting equation

$6y + 10 = 22$ (subtract 10 from both sides)
$6y = 12$ (divide both sides by 6)
$y = 2$

Remember to write down each operation you're using at every step. This helps avoid mistakes and shows your working clearly.

Equations with the letter on both sides

When the unknown letter appears on both sides of the equation, you need to collect all the letter terms together on one side before solving.

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Key process:

Start by collecting like terms so all letters are on one side
You can add or subtract multiples of the unknown from both sides
Always do the same operation to both sides to maintain balance
Solve the resulting single-sided equation

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Worked Example: Letter on Both Sides

Solve $4x + 26 = 2 - 2x$

Step 1: Collect x terms on one side

$4x + 26 = 2 - 2x$ (add $2x$ to both sides)
$6x + 26 = 2$ (subtract 26 from both sides)
$6x = -24$ (divide both sides by 6)
$x = -4$

Worked examples

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Worked Example 1: Combining Brackets and Both Sides

Solve $7p + 2 = 5(p - 4)$

First, expand the brackets on the right side: $7p + 2 = 5p - 20$

Then collect like terms by subtracting $5p$ from both sides:

$2p + 2 = -20$
$2p = -22$ (subtract 2 from both sides)
$p = -11$ (divide both sides by 2)

Checking your answer: Substitute $p = -11$ back into both sides:

Left side: $7(-11) + 2 = -77 + 2 = -75$ ✓
Right side: $5(-11 - 4) = 5(-15) = -75$ ✓

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Worked Example 2: Equations with Fractions

Solve $\frac{10 - y}{5} = 3$

First, multiply both sides by 5 to remove the fraction: $10 - y = 15$

Then solve for y:

$10 = 15 + y$ (add y to both sides)
$-5 = y$ (subtract 15 from both sides)

So y = -5. You can write this as $-5 = y$ or $y = -5$ - both are correct as long as the letter is isolated on one side.

Key strategies

The following strategies will help you tackle different types of linear equations effectively:

For equations with brackets:

Always expand brackets first
Then treat as a standard linear equation
Show each step clearly

For equations with letters on both sides:

Decide which side to collect the letter terms on
Usually choose the side that will give a positive coefficient
Move all letter terms to one side and number terms to the other
Remember: whatever you do to one side, do to the other

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For checking solutions:

Substitute your answer back into the original equation
Both sides should give the same value
This confirms your solution is correct

Remember!

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Key Points to Remember:

Always expand brackets first before attempting to solve the equation
Collect like terms by moving all letter terms to one side of the equation
Do the same operation to both sides to maintain the equation's balance
Work through one step at a time and write down each operation clearly
Check your answer by substituting back into the original equation

Linear equations 2 (AQA GCSE Maths): Revision Notes