Final answer:
The system of equations y = 2x^2 and y = -3x - 1 is solved using the substitution method.
We set the expressions equal to each other and solve the resulting quadratic equation to find the x-values.
The solutions to the system are (-1/2, 1/2) and (-1, 2).
Step-by-step explanation:
To solve the system of equations y = 2x^2 and y = -3x - 1, we will use the substitution method since both equations are already solved for y.
This implies setting the two expressions for y equal to each other and solving for x.
- Set 2x^2 equal to -3x - 1:
- 2x^2 = -3x - 1
- Rearrange the equation to form a quadratic equation:
- 2x^2 + 3x + 1 = 0
- Factor the quadratic equation (if possible) or use the quadratic formula to find the x-values.
- In this case, the equation factors to:
- (2x + 1)(x + 1) = 0
- Set each factor equal to zero and solve for x:
- x = -1/2 or x = -1
- Substitute the x-values back into one of the original equations to find the corresponding y-values:
- For x = -1/2:
- y = 2(-1/2)^2 = 1/2
- For x = -1:
- y = 2(-1)^2 = 2
- Thus, we have two solutions for the system:
- (-1/2, 1/2) and (-1, 2)