Deriving Simultaneous Equations

## Deriving Simultaneous Equations

Simultaneous equations involve two sets of variables. Deriving a simultaneous equation from a text involves:

• determining the two variables involved;

• identifying the multiples associated with each of the two variables;

• identifying the sum of each of the multiple + variable pairs.

## Example 1

At a garden centre, four shrubs and two trees cost £56. Five shrubs and one tree cost £52.

How much would an individual shrub cost?

 Let shrubs = s and trees = t: 4s + 2t = 56 and 5s + t = 52 Rearrange the second equation t = 52 - 5s Substitute t into the 1st equation: 4s+ 2(52- 5s) = 56 Expand the bracket: 4s+ 104- 10s = 56 Add the s together 104- 6s = 56 Add 6s to both sides 104 = 56+ 6s Subtract 56 from both sides 48 = 6s Divide both sides by 6 8 = s Substitute into 5s+t=52 5(8) + t=52 t=12 Check into 4s+2t=56 4(8) + 2(12) = 56 ✔

Answer: shrub = £8

## Example 2

Two families went to the same restaurant. The Khans had 3 pizzas and one pasta; the Smiths had 2 pizzas and 2 pastas. The bill for the Khans was £35.00, which was £1.50 more than the bill for the Roberstons.

How much was a pizza at the restaurant?

 Let x = pizza and y = pasta Robertsons: 3x + y = 35.00 - 1.50 Khans: 2x + 2y = 35.00 Rearrange 1st equation: y = 33.5 - 3x Second equation: 2x+ 2y = 35 Substitute into 2nd equation 2x+ 2(33.5- 3x) = 35 Expand the brackets 2x+ 67- 6x = 35 Add 4x to both sides 67 = 35+ 4x Subtract 35 from both sides 32 = 4x Divide both sides by 4 8 = x Substitute x=8 into 2nd equation: 2(8) + 2y=35 y=9.5 Check: 3(8) + 2(9.5) = 33.5 ✔