Question

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

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

Answer:

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

Step-by-step 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,

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