Solving Simultaneous Equations

# Solving Simultaneous Equations

GCSE(F), GCSE(H),

Simultaneous Equations involve two equations with two unknowns. To solve a simultaneous Equation:

• Rearrange one equation so that one unknown is on one side

• In the second equation, replace the unknown from the first equation

• Solve the second equation (with one unknown)

• Replace the solved value in the first equation

• Solve the first equation

• Check the values are correct

Note that a) it is not important which equation is used to start, and b) if one equation is a multiple of another, then there will be no solution (i.e. they are parallel lines).

## Examples

1. Solve the simultaneous equations 4x-y=6 text( and ) 5x+2y=1

Answer: x=1, y=-2

Rearrange the first equation into terms of y: y = 4x - 6

Substitute this into the second equation: 5x+2(4x-6)=1

And solve to obtain x=1

Substitute the known value of x into the first equation: 4(1)-y=6

And solve for y: y=-2

Finally, check both equations with known values: 4(1)-(-2)=6 (true) and 5(1)+2(-2)=1(true)

2. Solve the simultaneous equations 4x+3y=14 text( and ) 6x+2y=11

Answer: x=frac(1)(2) text( and ) y=4

Rearrange the second equation in terms of y: y=frac(11)(2)-3x

Substitute into the first equation: 4x+3(frac(11)(2)-3x)=14

And solve for x: x=frac(1)(2)

Substitute back into the second equation: 6(frac(1)(2))+2y=11

And solve for y: y=4

Check: 4(frac(1)(2))+3(4)=14 (true) and 6(frac(1)(2))+2(4)=11 (true)