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Q. 6

Expert-verified
Found in: Page 772

Precalculus Enhanced with Graphing Utilities

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

Graph each equation of the system. Then solve the system to find the points of intersection.$\left\{\begin{array}{l}y={x}^{2}+1\\ y=4x+1\end{array}\right\$

Solutions of system of equations$\left\{\begin{array}{l}y={x}^{2}+1\\ y=4x+1\end{array}\right\$ are $\left(0,1\right)&\left(4,17\right)$and the graph is

See the step by step solution

Step 1. Given data

The given system of equations is

$\left\{\begin{array}{l}y={x}^{2}+1\cdots \left(i\right)\\ y=4x+1\cdots \left(ii\right)\end{array}\right\$

Step 2. Graph of the system of equations

Plot the graph of $y={x}^{2}+1&y=4x+1$on same cartesian plane

Step 3. Solution of system

subtract equation ii from equation i

$y-y=\left({x}^{2}+1\right)-\left(4x+1\right)\phantom{\rule{0ex}{0ex}}0={x}^{2}-4x\phantom{\rule{0ex}{0ex}}0=x\left(x-4\right)$

By zero product property

$x-4=0\phantom{\rule{0ex}{0ex}}x=4$

or

$x=0$

So $x=0&4$

Step 4. Solution of system

Substitute $x=0$ in equation ii

role="math" localid="1646869492730" $y=4x+1\phantom{\rule{0ex}{0ex}}y=4\left(0\right)+1\phantom{\rule{0ex}{0ex}}y=1$

Substituterole="math" localid="1646869479187" $x=4$ in equation ii

role="math" localid="1646869502982" $y=4x+1\phantom{\rule{0ex}{0ex}}y=4\left(4\right)+1\phantom{\rule{0ex}{0ex}}y=17$

So solutions of the system are role="math" localid="1646869524607" $\left(0,1\right)&\left(4,17\right)$

Step 5. Verification

Substitute $x=0&y=1$in the system

$\left\{\begin{array}{l}1={0}^{2}+1\\ 1=4\left(0\right)+1\end{array}\right\\to \left\{\begin{array}{l}1=1\\ 1=1\end{array}\right\$

Substitute $x=4&y=17$ in system

$\left\{\begin{array}{l}17={\left(4\right)}^{2}+1\\ 17=4\left(4\right)+1\end{array}\right\\to \left\{\begin{array}{l}17=17\\ 17=17\end{array}\right\$

The system of equations is satisfied by both solutions

so solutions $\left(0,1\right)&\left(4,17\right)$ are correct

Step 6. Point of intersection

Locate the$\left(0,1\right)&\left(4,17\right)$ for point of interception in the graph of a system of equations