Q. 6 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

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

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

$$\begin{gathered} y-y=\left(x^2+1\right)-\left(4x+1\right) \\ 0=x^2-4x \\ 0=x(x-4) \end{gathered}$$

 By zero product property

$$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$$

or

$x=0$

So $x=0 \& 4$

Step 4. Solution of system

Substitute $x=0$ in equation ii

$$\begin{gathered} y=4x+1 \\ y=4\left(0\right)+1 \\ y=1 \end{gathered}$$

Substitute $x=4$ in equation ii

$$\begin{gathered} y=4x+1 \\ y=4\left(4\right)+1 \\ y=17 \end{gathered}$$

So solutions of the system are  $(0,1) \& (4,17)$

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 $(0,1) \& (4,17)$ are correct

Step 6. Point of intersection

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