Year 10: Solving simultaneous linear equations

This cheat sheet will guide you through solving pairs of linear equations that have the same solution. These are known as simultaneous equations.

The Goal

We want to find the values of x and y that satisfy BOTH equations at the same time.

Methods

1. Elimination Method

This method involves adding or subtracting the equations to eliminate one variable. Look for coefficients that are multiples of each other.

1. Example: 2x + y = 7 x + y = 3
2. Subtract the second equation from the first: (2x + y) - (x + y) = 7 - 3 => x = 4
3. Substitute x = 4 back into either equation to solve for y: 4 + y = 3 => y = -1
4. Solution: x = 4, y = -1

2. Substitution Method

Solve one equation for one variable (e.g., solve for x) and substitute that expression into the other equation.

1. Example: x + y = 5 2x - y = 1
2. Solve the first equation for y: y = 5 - x
3. Substitute this into the second equation: 2x - (5 - x) = 1
4. Simplify and solve for x: 2x - 5 = 1 => 2x = 6 => x = 3
5. Substitute x = 3 back into either equation to solve for y: 3 + y = 5 => y = 2
6. Solution: x = 3, y = 2

Key Tips

• Always check your solution by substituting the values of x and y back into both original equations.
• Be careful with signs (+ and -).