Please help with these 3 questions.
1. Suppose that y varies inversely with x, and y=3, when x=5. What is an equation fpr the inverse variation.
2. What are the solutions to the system?
y=x^2+2x-45
y=-3x+5
3. -sqrt 18 divided by sqrt 12
Thank you!
1. xy = 15
2. substituting
... -3x + 5 = x^2 + 2x - 45
... 0 = x^2 + 5x - 50
... factoring ... 0 = (x + 10)(x - 5)
... x = -10 , x = 5
substitute the x-values back to find the corresponding y-values
3. -√(18/12) = ±√(3/2)
1. To find the equation for inverse variation, you need to use the formula for inverse variation, which is y = k/x, where k is the constant of variation. In this case, we are given that when y = 3, x = 5, so we can substitute these values into the equation:
3 = k/5
To find the value of k, we can cross multiply:
3 * 5 = k
15 = k
Now that we have the value of k, we can rewrite the equation for inverse variation:
y = 15/x
So, the equation for the inverse variation is y = 15/x.
2. To find the solutions to the system of equations, we need to find the values of x and y that satisfy both equations simultaneously. Let's solve this system of equations:
First, the equations are:
y = x^2 + 2x - 45 (Equation 1)
y = -3x + 5 (Equation 2)
To solve this system, we can set the right sides of both equations equal to each other:
x^2 + 2x - 45 = -3x + 5
Now, let's rearrange the equation to get a quadratic equation in standard form:
x^2 + 2x + 3x - 45 - 5 = 0
x^2 + 5x - 50 = 0
Now, we can factor this quadratic equation:
(x + 10)(x - 5) = 0
Setting each factor equal to zero, we can solve for x:
x + 10 = 0 or x - 5 = 0
Solving each equation for x:
x = -10 or x = 5
Now that we have the values of x, we can substitute them back into either of the original equations to find the corresponding values of y. Let's use Equation 2:
For x = -10:
y = -3(-10) + 5
y = 30 + 5
y = 35
For x = 5:
y = -3(5) + 5
y = -15 + 5
y = -10
So, the solutions to the system of equations are (x = -10, y = 35) and (x = 5, y = -10).
3. To simplify the expression -sqrt(18)/sqrt(12), we can rationalize the denominator.
First, let's simplify the square roots:
sqrt(18) = sqrt(9 * 2) = sqrt(9) * sqrt(2) = 3 * sqrt(2)
sqrt(12) = sqrt(4 * 3) = sqrt(4) * sqrt(3) = 2 * sqrt(3)
Now, substitute the simplified square roots back into the expression:
(-3 * sqrt(2)) / (2 * sqrt(3))
To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator, which is (2 * sqrt(3)):
((-3 * sqrt(2)) / (2 * sqrt(3))) * ((2 * sqrt(3)) / (2 * sqrt(3)))
Simplifying further:
(-3 * sqrt(2) * 2 * sqrt(3)) / (2 * sqrt(3) * 2 * sqrt(3))
Now, cancel out the common factors:
(-6 * sqrt(2) * sqrt(3)) / (4 * 3)
(-6 * sqrt(6)) / 12
Simplifying the fraction:
-1/2 * sqrt(6)
So, the simplified expression is -1/2 * sqrt(6).