Chapter 9.7, Problem 2WE

To determine

To calculate: To find the number of real solutions of system of equations by sketching graphs

Answer to Problem 2WE

Number of real solutions is $2$

Explanation of Solution

Given information: System of equations $16x^2+4y^2=64;3x+4y=12$

Calculation:

Given the system of equations as follows:

  $\begin{array}{l}16x^2+4y^2=64----(1)\\ 3x+4y=12------(2)\end{array}$

When $x=0$

  $\begin{array}{l}16{\left(0\right)}^2+4y^2=64\\ 4y^2=64\\ y^2=16\\ y=\pm 4\end{array}$   And  $\begin{array}{l}3\left(0\right)+4y=12\\ 4y=12\\ y=3\end{array}$

When $y=0$

  $\begin{array}{l}16x^2+4{\left(0\right)}^2=64\\ 16x^2=64\\ x^2=4\\ x=\pm 2\end{array}$   And  $\begin{array}{l}3x+4\left(0\right)=12\\ 3x=12\\ x=4\end{array}$

Plotting the graph of equations on same plane:

  

From graph, the two equations intersects at two points

Conclusion:

Hence, number of real solutions is $2$