x - 2y = 1
Add 2y to both sides of the equation.
x = 1 + 2y
x2 - 3y2 = 13
Since x = 1 + 2y we can replace x with 1 + 2y .
(1 + 2y)2 - 3y2 = 13
Multiply out (1 + 2y)2 .
(1 + 2y)(1 + 2y) - 3y2 = 13
(1)(1) + (1)(2y) + (2y)(1) + (2y)(2y) - 3y2 = 13
1 + 2y + 2y + 4y2 - 3y2 = 13
1 + 4y + y2 = 13
y2 + 4y + 1 = 13
Subtract 13 from both sides.
y2 + 4y - 12 = 0
Factor the left side.
(y + 6)(y - 2) = 0
Set each factor equal to zero and solve for y .
y + 6 = 0 or y - 2 = 0
y = -6 or y = 2
Now let's use the equation x = 1 + 2y to find x .
If y = -6 then x = 1 + 2(-6) = 1 + -12 = -11
So one solution is x = -11 and y = -6 .
If y = 2 then x = 1 + 2(2) = 1 + 4 = 5
So another solution is x = 5 and y = 2 .
The solution set is { (-11, -6), (5, 2) }