solve the system algebraically and graphically 32x^2+9y^2=324 3y^2-x^2=3
32x^2+9y^2=324
3y^2-x^2=3
Since 3y^2 = x^2+3, we have
32x^2 + 3(x^2+3) = 324
I think you can solve that fairly easily, right?
The intersections are at x=±3, y=±2.
See the graphs at
http://www.wolframalpha.com/input/?i=plot+32x^2%2B9y^2%3D324%2C+3y^2-x^2%3D3
To solve the system of equations algebraically and graphically, we will solve for one variable in terms of the other in one equation and substitute that expression into the other equation. Let's solve the system step by step.
1. Algebraic Solution:
First, let's solve the second equation for x^2 in terms of y^2:
3y^2 - x^2 = 3
x^2 = 3y^2 - 3
Next, substitute x^2 in the first equation:
32x^2 + 9y^2 = 324
32(3y^2 - 3) + 9y^2 = 324
96y^2 - 96 + 9y^2 = 324
105y^2 = 420
y^2 = 420 / 105
y^2 = 4
y = ±√4
y = ±2
Substitute the values of y back into x^2 = 3y^2 - 3 to find x:
x^2 = 3(2^2) - 3
x^2 = 3(4) - 3
x^2 = 12 - 3
x^2 = 9
x = ±√9
x = ±3
Therefore, the algebraic solution to the system is (x, y) = (3, 2), (3, -2), (-3, 2), (-3, -2).
2. Graphical Solution:
To solve the system graphically, we will plot the given equations on a graph and look for their intersection points.
Graph the equation 32x^2 + 9y^2 = 324:
Divide both sides by 324 to simplify the equation:
x^2/9 + y^2/36 = 1
The graph of this equation is an ellipse centered at the origin, with the semi-major axis of length 6 (sqrt(36)) along the y-axis and the semi-minor axis of length 3 (sqrt(9)) along the x-axis.
Graph the equation 3y^2 - x^2 = 3:
Rearrange the equation to the standard form:
x^2 - 3y^2 = -3
The graph of this equation is a hyperbola centered at the origin, with the transverse axis along the x-axis.
Now, plot these two graphs on the same coordinate system. The intersection points of the two graphs will be the solutions to the system.
In this case, it turns out that the graphs do not intersect at any points in the real plane. Therefore, there are no real solutions to the system.
In summary, the algebraic solution to the system is (x, y) = (3, 2), (3, -2), (-3, 2), (-3, -2), and the graphical solution shows no intersection between the two equations.