Question

What are the solutions to the following system of equations?

-2x + y = -5
y = -3x² + 5

A) (0, 2)
B) (1, -2)
C) (12, -1) and (-13, -1)
D) (15, -10) and (-15, -10)

Answer 1

To find the solutions to the given system of equations, use the substitution method. Substitute the value of y from the second equation into the first equation and solve for x. Substitute the values of x back into the second equation to find the corresponding y-values.

Step-by-step explanation:

To find the solutions to the given system of equations, we can use substitution or elimination method. Let's use substitution method.

Given equations: -2x + y = -5 and y = -3x^2 + 5.

1. Substitute the value of y from the second equation into the first equation:
2. -2x + (-3x^2 + 5) = -5.
3. Simplify and solve for x:
4. -2x - 3x^2 + 5 = -5.
5. Rearrange the equation and set it equal to zero:
6. 3x^2 -2x - 10 = 0.
7. Now, factor the quadratic equation:
8. (3x - 10)(x + 1) = 0.
9. Set each factor equal to zero and solve for x:
10. 3x - 10 = 0 or x + 1 = 0.
11. Solve for x:
12. x = 10/3 or x = -1.
13. Substitute the values of x back into the second equation to find the corresponding y-values:
14. When x = 10/3, y = -3(10/3)^2 + 5 = 2.
15. When x = -1, y = -3(-1)^2 + 5 = -2.

Therefore, the solutions to the given system of equations are:
(10/3, 2) and (-1, -2).