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).

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)

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