Question

Solve the simultaneous equations `2x+y=8.......................................(1)` `4x^2+3y^2=52.......................(2)` ?

Answer 1

`x= 3.5 and y =1`
OR
`x = 2.5 and y = 3`

Explanation:

`2x+y=8.......................................(1)`

`4x^2+3y^2=52.......................(2)`

(1) `=> y=8-2x`

(2) `=> 4x^2+3(8-2x)^2=52`

`=> 4x^2 +3(64 - 32x +4x^2) =52`

`=> 4x^2 + 192 - 96x + 12x^2 = 52`

`=> 16x^2 -96x + 140 = 0`

`=> 4(4x^2 - 24x +35) = 0`

`=> 4x^2 -24x +35 = 0`
Solving this quadratic equation, we get:

`=> (x-3.5)(x-2.5) = 0`

`=> x= 3.5 or x = 2.5`

Substitute this value of `x` in equation (1):

Case 1: Taking `x= 3.5`

`=> 2x+y=8`

`=> 2(3.5) + y = 8`

`=> y = 8-7 =1`

OR

Case 2 : Taking `x= 2.5`

`2(2.5) + y = 8`

`=> y = 8- 5 = 3`

`therefore x= 3.5 and y =1`

OR

`x = 2.5 and y = 3`

Answer 2

`y=3 and x=5/2 or y=1 and x=7/2`

Explanation:

`2x+y=8`
`4x^2+3y^2=52`

`2x=8-y=>4x^2=(8-y)^2=color(blue)(64-16y+y^2)`

`color(blue)(64-16y+y^2)+3y^2=52`
`4y^2-16y+12=0`

dividing everything by 4(for symplifiying calculations):

`y^2-4y+3=0`

`y=(4+-sqrt(4^2-4*1*3))/2`

`y=(4+-sqrt(4))/2`

`y=(4+-2)/2`

`y=3 and x=5/2 or x=1 and x=7/2`