Simultaneous Equations Revision Notes for Edexcel GCSE Maths

Simultaneous equations

What are simultaneous equations?

Simultaneous equations are a pair of equations that must be solved together to find the values of two unknown variables (usually $x$ and $y$ ). You need to find values that satisfy both equations at the same time. There are two main types you'll encounter in your GCSE maths exam, and each requires a different approach.

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The word "simultaneous" means "at the same time" - so you're looking for values that work in both equations simultaneously.

Types of simultaneous equations

Easy simultaneous equations

These involve two linear equations where both unknowns appear to the first power only. For example:

$2x = 6 - 4y$
$-3 - 3y = 4x$

Tricky simultaneous equations

These involve one linear equation and one quadratic equation (where one unknown is squared). For example:

$7x + y = 1$
$2x² - y = 3$

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Key difference: Easy equations use the elimination method, while tricky equations require the substitution method because of the quadratic term.

Method 1: Six steps for easy simultaneous equations

When you have two linear equations, you can use the elimination method. This works by making the coefficients of one variable the same, then adding or subtracting the equations to eliminate that variable.

Step 1: Rearrange both equations Put both equations into the standard form $ax + by = c$ , where $a$ , $b$ , and $c$ are numbers (which can be positive or negative). Label your equations as ① and ②.

Step 2: Match up the coefficients Look at the numbers in front of either the $x$ 's or $y$ 's in both equations. You may need to multiply one or both equations by suitable numbers to make these coefficients match. Relabel your new equations as ③ and ④.

Step 3: Add or subtract the equations To eliminate the variable with matching coefficients, follow this rule:

If the coefficients have the same sign (both positive or both negative), subtract the equations
If the coefficients have opposite signs (one positive, one negative), add the equations

Step 4: Solve the resulting equation After elimination, you'll have a simple equation with just one unknown. Solve this to find the value of one variable.

Step 5: Substitute back Take the value you found and substitute it back into one of your original equations to find the other variable.

Step 6: Check your answer Substitute both values into the other original equation to verify your solution is correct. If it doesn't work, you've made an error and need to start again.

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Worked Example: Solving Linear Simultaneous Equations

Given: $2x + 3y = 7$ ① and $x - y = 1$ ②

Step 1: Already in standard form Step 2: Multiply equation ② by 2 to get $2x - 2y = 2$ ③ Step 3: Subtract ③ from ①: $(2x + 3y) - (2x - 2y) = 7 - 2$ This gives: $5y = 5$ , so $y = 1$ Step 4: Substitute $y = 1$ into ②: $x - 1 = 1$ , so $x = 2$ Step 5: Check in ①: $2(2) + 3(1) = 4 + 3 = 7$ ✓

Method 2: Seven steps for tricky simultaneous equations

When you have one linear and one quadratic equation, you need to use substitution rather than elimination. This is because the quadratic term makes elimination much more complicated.

Step 1: Rearrange the linear equation Rearrange the linear equation so that one unknown is on its own on one side. Label your equations as ① and ②.

Step 2: Substitute the expression Take the expression for the isolated unknown and substitute it into the quadratic equation. This will give you a new equation - label it ③.

Step 3: Rearrange to get a quadratic equation Expand and rearrange your new equation to get it in the standard quadratic form. You'll need to solve this quadratic equation.

Step 4: Solve the quadratic Use factorisation, completing the square, or the quadratic formula to find the values of your variable. You should get two solutions.

Step 5: Find the corresponding values Substitute each solution back into one of your original equations (choose the easier one) to find the corresponding values of the other variable.

Step 6: Check both pairs Substitute both pairs of values into the other original equation to verify they work.

Step 7: Write out your final answer Clearly state both pairs of solutions, as there will typically be two valid answers for this type of problem.

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Worked Example: Solving Mixed Simultaneous Equations

Given: $y = 2x + 1$ ① and $x² + y² = 10$ ②

Step 1: Equation ① is already rearranged Step 2: Substitute into ②: $x² + (2x + 1)² = 10$ Step 3: Expand: $x² + 4x² + 4x + 1 = 10$ Simplify: $5x² + 4x - 9 = 0$ Step 4: Factor: $(5x + 9)(x - 1) = 0$ So $x = -\frac{9}{5}$ or $x = 1$ Step 5: When $x = 1$ : $y = 2(1) + 1 = 3$ When $x = -\frac{9}{5}$ : $y = 2(-\frac{9}{5}) + 1 = -\frac{13}{5}$ Step 6: Check both solutions in ②

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Remember that tricky simultaneous equations will usually give you two different solutions because of the quadratic nature of one equation.

Important tips

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Common Mistakes to Avoid:

Always check your solutions by substituting back into both original equations
For tricky equations, remember you'll get two pairs of solutions, not just one
When matching coefficients in easy equations, you might need to multiply by negative numbers
Take your time with the arithmetic - small errors can lead to completely wrong answers
If your check doesn't work, don't panic - just go back through your working to find the mistake

The key function of checking your answer cannot be overstated - it's essential for catching calculation errors before you submit your work.

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Key Points to Remember:

Two types exist: Easy equations (both linear) use elimination, tricky equations (one linear, one quadratic) use substitution
Always rearrange first: Get equations into standard form before starting your method
Check your work: Substitute your final answers back into both original equations to verify they're correct
Tricky equations give two solutions: Don't forget to find both pairs of values when dealing with quadratic equations
Follow the steps systematically: These methods work when you follow them carefully in the right order

Simultaneous Equations (Edexcel GCSE Maths): Revision Notes