Question

Make and explain a prediction about the nature of the solution to the following system of equations, and then solve the system. x²+y²=25, 4x+3y=0 Illustrate with a graph. Verify your solution, and compare it with your initial prediction.

Answer 1

Answer:

(3,-4) and (-3,4)

Step-by-step explanation:

$x^2+y^2=25$

$4x+3y=0$

The first equation is a circle with radius of 5 units

The second equation is a line which has slope $\frac{4}{3}$

So, for 4 unit change in y there will be 3 unit chang in x.

The circle has its center at the origin and the line passes through the orgin.

The line will intersect the circle at two points giving us two solutions

From the second equation we get

$y=-\frac{4}{3}x$

Applying in the second equation

$x^2+\left(-\frac{4}{3}x\right)^2=25\\\Rightarrow x^2+\frac{16}{9}x^2=25\\\Rightarrow \frac{9x^2+16x^2}{9}=25\\\Rightarrow 25x^2=225\\\Rightarrow x^2=\frac{225}{25}\\\Rightarrow x^2=9\\\Rightarrow x=\pm 3$

When x = 3

$y=-\frac{4}{3}x\\\Rightarrow y=-\frac{4}{3}\times 3\\\Rightarrow y=-4$

When x = -3

$y=-\frac{4}{3}x\\\Rightarrow y=-\frac{4}{3}\times -3\\\Rightarrow y=4$

So, the points where the line and circle intersect are (3,-4) and (-3,4)