Answer:
To solve this system of equations:
1. Start with the first equation:
x - 6y = -6
2. Rearrange it to express x in terms of y:
x = 6y - 6
3. Substitute this expression for x into the second equation:
2(6y - 6)^2 + y^2 = 76
4. Simplify and solve for y:
2(36y^2 - 72y + 36) + y^2 = 76
72y^2 - 144y + 72 + y^2 = 76
73y^2 - 144y + 72 = 76
73y^2 - 144y - 4 = 0
5. You now have a quadratic equation in terms of y. You can solve this quadratic equation using the quadratic formula:
y = [144 ± sqrt(144^2 - 4 * 73 * (-4))] / (2 * 73)
Calculate the values of y using the quadratic formula:
y1 ≈ 2.159
y2 ≈ 0.018
6. Now that you have two potential values for y, you can use the expression for x to find the corresponding values of x:
For y = 2.159:
x = 6 * 2.159 - 6 ≈ 11.954
For y = 0.018:
x = 6 * 0.018 - 6 ≈ -5.892
So, the solutions to the system of equations are approximately:
(x1, y1) ≈ (11.954, 2.159)
(x2, y2) ≈ (-5.892, 0.018)