Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Solve the system of equations algebraically. Show all of your steps.
y=x^2+2x
y=3x+20
To solve this system of equations, we will set the two equations equal to each other.
x^2 + 2x = 3x + 20
Subtracting 3x and 20 from both sides:
x^2 - x - 20 = 0
Now we need to factor the quadratic equation:
(x - 5)(x + 4) = 0
Setting each factor to zero:
x - 5 = 0 or x + 4 = 0
x = 5 or x = -4
Now substitute these values back into one of the original equations to solve for y:
For x = 5:
y = 5^2 + 2(5) = 25 + 10 = 35
So one solution is (5, 35).
For x = -4:
y = (-4)^2 + 2(-4) = 16 - 8 = 8
So the other solution is (-4, 8).
Therefore, the system of equations has two solutions: (5, 35) and (-4, 8).