Deriving Simultaneous Equations

Deriving Simultaneous Equations

GCSE(F), GCSE(H),

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.

Examples

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 and tree cost?

Answer: shrub = £8, tree = £12

Let shrubs = s and trees = t

`text([1] )4s + 2t = 56` and

`text([2] )5s + t = 52`

Rearrange the second equation:

`t = 52 - 5s`

Substitute into the first equation:

`4s + 2(52 - 5s) = 56`

`4s + 104 - 10s = 56`

`6s = 48` giving

`s = 8`

Substitute back:

`5(8) + t = 52` gives `t=12`

Check: `4(8)+2(12)=56`

2. Two families went to the same restaurant. The Roberstons 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?

Answer: £8.00

Let `x = text(pizza) and y = text (pasta)`

Robertsons: `3x + y = 33.50`

Smiths: `2x + 2y = 35.00`

Rearranging the first equation: `y = 33.5 - 3x`

Substitute into the second equation: `2x + 2(33.5 - 3x) = 35`

Rearrange and solve for `x=8` (the price of the pizza)

Substitute into the second equation to obtain the value for the pasta:

`2xx8 + 2y = 35; 2y=29; y=9.5`

Check with equation 1: `3xx8 + 2xx9.5=33.5`

(Solve for both variables and carry out the check to ensure that the answer given is correct.)