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Q. 8

Expert-verified
Found in: Page 772

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

In Problems 5–24, graph each equation of the system. Then solve the system to find the points of intersection.$y=\sqrt{4-{x}^{2}};\phantom{\rule{0ex}{0ex}}y=2x+4$

he graph of the system of equations $y=\sqrt{4-{x}^{2}};y=2x+4$ is:

The points of intersection are $\left(-\frac{6}{5},\frac{8}{5}\right),\left(-2,0\right)$

See the step by step solution

Step 1. Given

The system of non-linear equation:

$y=\sqrt{4-{x}^{2}};\phantom{\rule{0ex}{0ex}}y=2x+4$

To graph the equation and to find the point of intersection.

Step 2. Graph the equations

Graph the equations in the same plane.

Step 3. To find the point of intersection.

Equate both the equations,

$\sqrt{4-{x}^{2}}=2x+4$

Squaring on both sides, we get,

$4-{x}^{2}={\left(2x+4\right)}^{2}\phantom{\rule{0ex}{0ex}}4-{x}^{2}=4{x}^{2}+16x+16\phantom{\rule{0ex}{0ex}}5{x}^{2}+16x+12=0$

Step 4. Solve the equations

$5{x}^{2}+16x+12=0\phantom{\rule{0ex}{0ex}}\left(5x+6\right)\left(x+2\right)=0\phantom{\rule{0ex}{0ex}}x=-\frac{6}{5},-2$

The value of $x$ is $-\frac{6}{5},-2$

Step 5. Find y

When

$x=-\frac{6}{5},\phantom{\rule{0ex}{0ex}}y=2x+4\phantom{\rule{0ex}{0ex}}=2\left(-\frac{6}{5}\right)+4\phantom{\rule{0ex}{0ex}}=-\frac{12}{5}+4\phantom{\rule{0ex}{0ex}}=\frac{8}{5}$

When $x=-2,$

$y=2x+4\phantom{\rule{0ex}{0ex}}=2\left(-2\right)+4\phantom{\rule{0ex}{0ex}}=0$

The point of intersections are $\left(-\frac{6}{5},\frac{8}{5}\right),\left(-2,0\right)$