Question

How to solve the following simultaneous equations: `2x+y=8` `4x^2+3y^2=52` ??

Answer 1

Write the linear equation as y in terms of x.
Substitute for y into the elliptic equation.
Solve the resulting quadratic for the values of x.
Use the linear equation to find the y values.

Explanation:

Write the linear equation as y in terms of x.

`y = 8-2x`

Substitute for y into the elliptic equation.

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

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

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

Solve the resulting quadratic for the values of x.

`x = (96+sqrt((-96)^2-4(16)(140)))/(2(16))`

and

`x = (96-sqrt((-96)^2-4(16)(140)))/(2(16))`

`x = 7/2` and `x = 5/2`

Use the linear equation to find the y values.

`y = 8-2(7/2)` and `y = 8-2(5/2)`

`y = 1` and `y = 3`

The points of intersection for the line and the ellipse are `(7/2,1)` and `(5/2,3)`