Question

Barbie is analyzing a circle, y2 + x2 = 16, and a linear function g(x). Will they intersect?

y2 + x2 = 16 g(x)

graph of the function y squared plus x squared equals 16

x g(x)

0 6

1 3

2 0

Answer 1

we have that
y2 + x2 = 16
and g(x)

using a graph tool
see the attached figure
the graphs intersect at two points
therefore
the system has two solutions

Answer 2

They will intersect twice, at x = 2.91 and x = 0.69.

Explanation:

First we write the function g(x).

From the table of values, we can see that for every increase of 1 in the value of x, the value of y decreases by 3. This makes the slope -3.

The y-intercept, the point where the data crosses the y-axis, will have an x-coordinate of 0. This makes (0, 6) our y-intercept.

This makes the equation of g(x), in slope-intercept form,

g(x) = -3x+6

This can also be written as y=-3x+6.

We will substitute this in place of y in our equation for the circle:

(-3x+6)²+x² = 16

(-3x+6)(-3x+6)+x²=16

-3x(-3x)+6(-3x)+-3x(6)+6(6)+x² = 16

9x²-18x-18x+36+x² = 16

10x²-36x+36 = 16

Subtracting 16 from each side,

10x²-36x+20 = 0

Using the quadratic formula,

`x=(-b\pm √(b^2-4ac))/(2a)\\\\=(--36\pm √((-36)^2-4(10)(20)))/(2(10)\\\\=(36\pm √(1296-800))/(20)\\\\=(36\pm √(496))/(20)\\\\=(36\pm 22.27)/(20)\\\\=(36+22.27)/(20)\text{ or }(36-22.27)/(20)\\\\=(58.27)/(20)\text{ or }(13.73)/(20)\\\\=2.91\text{ or }0.69`