Solving Simultaneous Equations - Substituton

## Solving Simultaneous Equations - Substituton

There are two ways to handle simultaneous equations.

This is the addition (or subtraction) method. Simultaneous Equations involve two equations with two unknowns. The substitution method involves:

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

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

• Solve this second equation (with one unknown)

• Replace the solved value in the first equation

• Solve the first equation

• Check the values are correct

Note that it is not important which equation is used to start.

If one equation is a multiple of another, then there will be no solution (i.e. they are parallel lines).

## Example 1

Solve the simultaneous equations 4x-y=6 and 5x+2y=1

Rearrange the first equation to get it in terms of y

 4x - y = 6 Add y to both sides: 4x = 6 + y Subtract 6 from both sides: 4x - 6 = y

Replace y in the second equation 4x-6

 Second equation: 5x + 2y = 1 substitute for y: 5x + 2(4x-6) = 1 expand brackets: 5x + 8x - 12 = 1 add x terms: 13x - 12 = 1 add 12 to both sides: 13x = 13 divide both sides by 13 x = 1

Replace x in the first equation with the found value of x

 First equation: 4x - y = 6 substitute for x: 4(1) - y = 6 4 - y = 6 add y to both sides: 4 = 6 + y subtract 6 from both sides: -2 = y

Finally, check by putting both values in the other equation

 Check using 2nd equation: 5(1)+2(-2)=1✔

Answer: x=1, y=-2

## Example 2

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

Rearrange the first equation to be in terms of y

 2nd equation to get y 6x + 2y = 11 subtract 6x from both sides: 2y = 11 - 6x divide all terms by 2 y = frac(11)(2) - 3x

Solve the equation for x

 First equation: 4x + 3y = 14 Replace the value y: 4x + 3(frac(11)(2)-3x) = 14 Multiply out the brackets: 4x + frac(33)(2) - 9x = 14 subtract the fraction from both sides: 4x   - 9x = -frac(5)(2) add the x-values: -5x     = -frac(5)(2) divide both sides by 5: -x     = -frac(1)(2) multiply both sides by -1: x     = frac(1)(2)

Substitute the found value of x

 Second equation 6x + 2y = 11 Substitute known value of x 6(frac(1)(2)) + 2y = 11 3 + 2y = 11 Subtract 3 from both sides   2y = 8 Divide both sides by 2   y = 4

 Check using 1st equation 4(frac(1)(2))+3(4)=14 ✔
Answer: x=frac(1)(2) and y=4`