Answer:
(x, y) = (-2, -1) or (1, 2)
Explanation:
You want to find the algebraic solutions to the system of equations ...
- f(x) = x² +2x -1
- g(x) = x +1
Solution
The x-value of the solutions will be the solutions to ...
f(x) = g(x)
f(x) -g(x) = 0
(x² +2x -1) -(x +1) = 0 . . . . substitute for f(x) and g(x)
x² +x -2 = 0 . . . . . . . . . simplify
(x +2)(x -1) = 0 . . . . . factor
The zero product rule says the solutions will be values of x that make one or the other of the factors zero.
x = -2 or +1 . . . . . . . values that make the factors zero
y = x +1 = -1 or +2 . . . . from the equation for g(x)
Solutions are (x, y) = (-2, -1) or (1, 2).
Proof
We already know that g(x) is satisfied by these x- and y-values.
f(x) = x² +2x -1 = (x +2)x -1
f(-2) = (-2 +2)(-2) -1 = 0 -1 = -1 . . . . . (-2, -1) is a solution
f(1) = (1 +2)(1) -1 = 3 -1 = 2 . . . . . . . . . (1, 2) is a solution
These values agree with the above, so we have shown the solutions satisfy both equations in the system of equations.