# 1)What is the 4x^2=y^2+8y+32

2)The graph of which equation is a circle?

3)Solve the system of equations by graphing x^2+y^2=16 and y= -x+4

4)Find the exact solutions of each system of equations x^2+y^2=25 and 9y=4x^2

5)Find the exact solutions of each system of equations y=x^2+1 and y=2x

#1 is a hyperbola.
The ?x^2 - ?y^2 .... is the clue.

5x^2+10x+5y=9 is a parabola

#3
solutions are (4.0) and (0,4)
use substitition
x^2 + (-x+4)^2 = 16
etc

#4
I got (3,4) and (-3,-4), they work in both equations, yours dont.
I multiplied the circle equation by 4 to get
4x^2 + 4y^2 = 100, then replaced 4x^2=9 into that, etc

#5
yes

## Here's the step-by-step explanation for each question:

1) To determine whether the equation 4x^2 = y^2 + 8y + 32 represents a hyperbola or a parabola, you can look at the form of the equation. The standard form of a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1, while the standard form of a parabola is y = ax^2 + bx + c. In this case, the given equation does not match either of these forms, so it is neither a hyperbola nor a parabola.

2) The equation 5x^2 + 10x + 5y = 9 can be rewritten as 5x^2 + 10x + 5y - 9 = 0. To determine if it represents a circle, you can look at the coefficients of x^2 and y^2. If they have the same value and are positive, the equation represents a circle. In this case, the coefficients are both 5, and since they are positive, the given equation represents a circle.

3) To solve the system of equations x^2 + y^2 = 16 and y = -x + 4 by graphing, you can plot the two equations on the same graph and find the points of intersection. The first equation represents a circle with a radius of 4, centered at the origin (0,0). The second equation represents a line with a slope of -1 and a y-intercept of 4. By graphing these two equations, you will find that the points of intersection are (4,0) and (0,-4), which are the solutions to the system of equations.

4) To find the exact solutions of the system of equations x^2 + y^2 = 25 and 9y = 4x^2, you can substitute one equation into the other to eliminate one variable. In this case, you can substitute 9y for 4x^2 in the first equation, resulting in 9y + y^2 = 25. This equation can be rearranged as y^2 + 9y - 25 = 0. By solving this quadratic equation, you will find the solutions y = 4 and y = -5. Substituting these values back into the equation 9y = 4x^2, you will find the corresponding x-values x = 3 and x = -3. Therefore, the exact solutions to the system of equations are (3,4) and (-3,-4).

5) To find the exact solutions of the system of equations y = x^2 + 1 and y = 2x, you can set the two equations equal to each other and solve for x. By substituting the value of y from the second equation into the first equation, you get x^2 + 1 = 2x. Rearranging this equation, you will find x^2 - 2x + 1 = 0. By solving this quadratic equation, you will find x = 1. Substituting this value back into the second equation, you will find y = 2. Therefore, the exact solution to the system of equations is (1,2).