To solve the given system of equations algebraically, we will first substitute the expression for y from the second equation into the first equation. This will allow us to solve for the value of x.
1. Start with the second equation: y = x + 8
2. Substitute this expression for y into the first equation: x + 8 = x² + 18x + 78
3. Rearrange the equation by moving all terms to one side to set it to zero: x² + 17x + 70 = 0
4. Factor the quadratic equation: (x + 7)(x + 10) = 0
5. Set each factor to zero and solve for x:
- x + 7 = 0 -> x = -7
- x + 10 = 0 -> x = -10
Now that we have found the values of x, we can substitute them back into either of the original equations to find the corresponding values of y.
For x = -7:
y = -7 + 8
y = 1
For x = -10:
y = -10 + 8
y = -2
Therefore, the solutions to the system of equations are:
x = -7, y = 1
x = -10, y = -2
