Question

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.

Answer 1

To use the substitution method, we need to solve one equation for one variable and substitute it into the other equation.

First, let's solve the first equation, y = x^2 - 9x + 39, for y:

y = x^2 - 9x + 39

Next, let's substitute this value for y into the second equation:

x^2 - 9x + 39 = 9x - 42

Now, let's solve this quadratic equation for x:

x^2 - 9x + 39 - 9x + 42 = 0

x^2 - 18x + 81 = 0

(x - 9)(x - 9) = 0

x - 9 = 0
x = 9

Now, let's substitute this value for x into one of the original equations to find y:

y = (9)^2 - 9(9) + 39

y = 81 - 81 + 39

y = 39

Therefore, the linear-quadratic system y = x^2 - 9x + 39, and y = 9x - 42 has one solution, (9, 39).