Question

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

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

Short answer

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

Step-by-step solution

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

$$\begin{gathered} Consider, \\ {y}^2-x^2=21.....\left(1\right) \\ y+x=7.....\left(2\right) \\ weget \\ {(7-x)}^2=21+x^2 \\ 49+x^2-14x=21+x^2 \\ x=2 \\ Nowsubstitutingvalueofxineq.2weget \\ y=7-2 \\ y=5 \end{gathered}$$

Step 4. Check the solution

$$\begin{gathered} Substitutetheobtainedvaluesofxandyinboththeequations \\ {y}^2-x^2=21 \\ 21=25-4 \\ 21=21 \\ also \\ x+y=7 \\ 7=5+2 \\ 7=7 \\ holdstrue. \end{gathered}$$