Final answer:
The solution to the system of equations y - x = 3 and x² - 7y + 31 = 0 is found by solving the first equation for y and substituting it into the second. Factoring the resulting quadratic equation gives us the solutions for x, which then give us the corresponding y values. The solution set is {(2, 5), (5, 8)}.
Step-by-step explanation:
To find the solution set for the system of equations y - x = 3 and x² - 7y + 31 = 0, we will solve the first equation for one variable and substitute it into the second equation. First, we solve for y in the first equation:
y = x + 3.
Next, we substitute y into the second equation:
x² - 7(x + 3) + 31 = 0
x² - 7x - 21 + 31 = 0
x² - 7x + 10 = 0.
This quadratic equation can be factored as:
(x - 2)(x - 5) = 0,
which gives us the solutions x = 2 and x = 5. Substituting these back into the first equation, we get the corresponding y values:
- x = 2 implies y = 2 + 3 = 5
- x = 5 implies y = 5 + 3 = 8
Therefore, the solution set is {(2, 5), (5, 8)}.