Q. 26 - Precalculus Enhanced with Graphing Utilities

Question

graph each equation of the system. Then solve the system to find the points of intersection.

$\left\{\begin{array}{l} x^2 -y^2=21\\ x+y=7\end{array}\right.$

Step-by-Step Solution

Answer

the point of intersection is $(5,2)$.

Step 1. Given

$\left\{\begin{array}{l} x^2 -y^2=21\\ x+y=7\end{array}\right.$

Step 2. Graphing each equation

$x^2 -y^2=21$

$x+y=7$

Step 3. Calculation of intersection point

$$\begin{gathered} Consider, \\ {y}^2- x^2 =21 .....\left(1\right) \\ y +x=7 .....\left(2\right) \\ we get \\ {(7-x)}^2=21+x^2 \\ 49+x^2-14x=21+x^2 \\ x=2 \\ Now substituting value of x in eq. 2 we get \\ y=7-2 \\ y=5 \end{gathered}$$

Step 4. Check the solution

$$\begin{gathered} Substitute the obtained values of x and y in both the equations \\ {y}^2 -x^2 =21 \\ 21=25-4 \\ 21=21 \\ also \\ x+y=7 \\ 7=5+2 \\ 7=7 \\ holds true. \end{gathered}$$