Q. 31 - Precalculus Enhanced with Graphing Utilities

Question

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

$\left\{\begin{array}{l}4x^2-3xy+9y^2=15\\ 2x+3y=5\end{array}\right.$

Step-by-Step Solution

Answer

The point of intersection is $\left(2,\frac{1}{3}\right) \& \left(\frac{1}{2},\frac{4}{3}\right)$.

Step 1. Given information

$\left\{\begin{array}{l}4x^2-3xy+9y^2=15\\ 2x+3y=5\end{array}\right.$

Step 2. Graphing each equation

$4x^2-3xy+9y^2=15$

$2x+3y=5$

Step 3. Calculation of intersection point

$$\begin{gathered} Consider, \\ 4x^2 - 3xy+9y^2=15 .....\left(1\right) \\ 2x+3y=5 .....\left(2\right) \\ Now substituting the value of y from eq.2 into eq.1 \\ we get \\ 4x^2-3x\left(\frac{5-2x}{3}\right)+9{\left(\frac{5-2x}{3}\right)}^2=15 \\ x=\left(2,\frac{1}{2}\right) \\ Now substituting value of x in eq. 2 we get \\ 2x+3y=5 \\ y=\left(\frac{1}{3},\frac{4}{3}\right) \end{gathered}$$