−3x − 8y = 20 −5x + y = 19 solved by substitution
2 answers:
Delete as this is the wrong answer. Made a typo in the answer.
Answer:
x =
y = 24
Step-by-step explanation:
-3x - 8y = 20 .... (1)
-5x + y = 19 .... (2)
What we can do is take the (2) equation
-5x + y = 19
and solve for y by adding 5x to both sides
y = 24
now that we have y we substitute it in for y in equation (1)
-3x - 8(24) = 20
-3x - 192 = 20
we want to solve for x so we need to get its self.
add 192 to both sides
-3x = 212
now we divide both sides by -3. It's going to be a fraction.
x =
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Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)