Simultaneous equations Revision Notes for Edexcel GCSE Maths

Simultaneous equations

What are simultaneous equations?

Simultaneous equations are pairs of equations that contain two unknown variables. Your job is to find the values of both unknowns that make both equations true at the same time.

For example:

$3x + y = 20$
$x + 4y = 14$

You need to find values of $x$ and $y$ that work in both equations.

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The key insight with simultaneous equations is that you're looking for the point where both mathematical relationships are satisfied simultaneously. Think of it as finding the common solution that makes both equations "happy" at the same time.

Algebraic solution method

This method uses elimination and substitution to solve the equations step by step. The algebraic approach gives you exact answers and works systematically through a proven process.

The five-step process

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The Five-Step Algebraic Method:

Step 1: Number each equation (①, ②)

Step 2: If needed, multiply one or both equations to make the coefficients of one unknown the same

Step 3: Add or subtract the equations to eliminate one unknown

Step 4: Once you find one unknown, substitute this value back into either original equation to find the other unknown

Step 5: Check your answer by substituting both values into the original equations

Worked example

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Worked Example: Solving Using Elimination

Let's solve: $6x + 2y = -3$ and $4x - 3y = 11$

Step 1: Number the equations

$6x + 2y = -3$ ①
$4x - 3y = 11$ ②

Step 2: Multiply to get matching coefficients

Multiply ① by 3: $18x + 6y = -9$
Multiply ② by 2: $8x - 6y = 22$

Step 3: Add the equations (the $6y$ terms cancel out)

$18x + 6y = -9$
$8x - 6y = 22$
Adding: $26x = 13$
So $x = \frac{1}{2}$

Step 4: Substitute $x = \frac{1}{2}$ into equation ①

$6(\frac{1}{2}) + 2y = -3$
$3 + 2y = -3$
$2y = -6$
$y = -3$

Step 5: Check by substituting into the equation you didn't use

$4(\frac{1}{2}) - 3(-3) = 2 + 9 = 11$ ✓

Solution: x = ½, y = -3

Tips for easier eliminations

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Time-Saving Shortcuts:

Look for these patterns to make elimination easier:

1. If an unknown appears on its own in one equation (like $x = 5$ ), you only need to multiply one equation to eliminate it

2. If an unknown has different signs in the two equations (like $+3x$ and $-3x$ ), you can eliminate it by adding the equations together

3. Always look for the smallest multipliers - this reduces the chance of arithmetic errors

Graphical solution method

You can solve simultaneous equations by drawing graphs of both equations on the same coordinate grid. This visual approach helps you understand what's happening mathematically.

The point of intersection gives you the solution - the $x$ and $y$ coordinates where the lines cross are your answers.

Example

For the equations $x - y = 1$ and $x + 2y = 4$ :

Draw both lines on a coordinate grid
Find where they intersect
The intersection point gives the solution: $x = 2$ , $y = 1$

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The graphical method is excellent for visualising the problem and understanding the geometric relationship between equations. However, the algebraic method is usually more accurate for finding exact answers, especially when solutions involve fractions or decimals.

Practice problems

Try solving these simultaneous equations using the methods you've learned:

Problem 1: (Algebraic method)

$3x - 2y = 12$
$x + 4y = 11$

Problem 2: (Graphical method)

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

Draw a coordinate grid from -3 to 5 in both directions to solve graphically.

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Key Points to Remember:

Simultaneous equations have two unknowns that you need to find
The algebraic method uses elimination and substitution in five clear steps
Always multiply equations carefully to get matching coefficients
Check your answer by substituting both values into an original equation
The graphical method finds the solution at the intersection point of two lines
Choose the algebraic method for exact answers, and the graphical method for visual understanding

Simultaneous equations (Edexcel GCSE Maths): Revision Notes