Q. 28 - Precalculus Enhanced with Graphing Utilities

Question

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

$\left\{\begin{array}{l}{x}^2-4y^2=16\\ 2y-x=2\end{array}\right.$

Step-by-Step Solution

Answer

The point of intersection is $\left(-5,-\frac{3}{2}\right)$.

Step 1. Given information

$\left\{\begin{array}{l}{x}^2-4y^2=16\\ 2y-x=2\end{array}\right.$

Step 2. Graphing each equation

$x^2-4y^2=16$

$2y-x=2$

Step 3. Calculation of intersection point

$$\begin{gathered} Consider, \\ {x}^2-4y^2=16 .....\left(1\right) \\ 2y-x=2 .....\left(2\right) \\ Now substituting the value of x from eq.2 into eq.1 \\ we get \\ {(2y-2)}^2-4y^2=16 \\ 4y^2+4-8y-4y^2=16 \\ y=-\frac{3}{2} \\ Now substituting value of y in eq. 2 we get \\ 2\left(-\frac{3}{2}\right)-2=x \\ x=-5 \end{gathered}$$