Solve the system of equations by substitution. (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.)
y = x^2 - 3x - 7
y = -2x^2 - 4x - 3
To solve the system of equations by substitution, we need to set the two expressions for y equal to each other.
So we have:
x^2 - 3x - 7 = -2x^2 - 4x - 3
Next, we can combine like terms by moving all the terms to one side of the equation:
3x^2 - x - 4 = 0
Now, we can solve this quadratic equation. However, to simplify the calculation, let's factorize the equation:
(3x + 4)(x - 1) = 0
Setting each factor equal to zero gives us two possible solutions:
3x + 4 = 0 or x - 1 = 0
Solving the first equation, we have:
3x = -4
x = -4/3
Solving the second equation, we have:
x = 1
So the possible solutions for the system of equations are x = -4/3 or x = 1.
Now, let's substitute these values of x back into one of the original equations (let's use the first one):
y = (1)^2 - 3(1) - 7
y = 1 - 3 - 7
y = -9
Therefore, the solutions to the system of equations are (-4/3, -9) and (1, -9).