determine the value of x and y if they satisfy both the following equations simultaneously:

Question

2x + y = 3
x^2 + y + x = y^2

Answer 1

Given equations:
1) 2x + y = 3
2) x^2 + y + x = y^2

Rewrite equation 1 in terms of y:
y = 3 - 2x

Substitute y = 3 - 2x into equation 2:
x^2 + (3 - 2x) + x = (3 - 2x)^2
x^2 + 3 - 2x + x = 9 - 12x + 4x^2
x^2 + x + 3 - 2x = 9 - 12x + 4x^2
x^2 + x + 3 - 2x = 9 - 12x + 4x^2
x^2 + x + 3 - 2x = 9 - 12x + 4x^2

Rearrange the terms:
x^2 + x + 3 - 2x = 9 - 12x + 4x^2
4x^2 - x - 6 = 0

Solving the above quadratic equation by factorization or using the quadratic formula:
(4x + 3)(x - 2) = 0
x = -3/4 or x = 2

Substitute x = -3/4 into equation 1:
2(-3/4) + y = 3
-3/2 + y = 3
y = 3 + 3/2
y = 9/2

Therefore, the solution to the system of equations is x = -3/4 and y = 9/2.

Similarly, for x = 2:
2(2) + y = 3
4 + y = 3
y = 3 - 4
y = -1

Therefore, the other solution to the system of equations is x = 2 and y = -1.