Use the substitution method to determine whether the linear-quadratic system y=x^2−9x+39, y=9x−42 has 0, 1, or 2 solutions.
To use the substitution method, we need to substitute one equation into the other. Let's substitute y=9x-42 into the first equation:
x^2 - 9x + 39 = 9x - 42
Simplifying this equation gives us:
x^2 - 18x + 81 = 0
This is a quadratic equation. We can either factor it or use the quadratic formula to solve for x.
If we factor the equation, we have:
(x - 9)(x - 9) = 0
This equation has a repeated factor, which means it only has one solution: x = 9.
Now let's substitute this value of x back into one of the original equations to find the corresponding value of y. Using y = 9x - 42:
y = 9(9) - 42
y = 81 - 42
y = 39
Therefore, the linear-quadratic system y = x^2 - 9x + 39 and y = 9x - 42 has one solution, which is (9, 39).