Chapter 9.7, Problem 9WE

To determine

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

Answer to Problem 9WE

Number of real solutions is $3$

Explanation of Solution

Given information: System of equations $x^2-4y^2=4;6y^2-x=2$

Calculation:

Given the system of equations as follows:

  $\begin{array}{l}{x}^2-4y^2=4-----(1)\\ 6y^2-x=2------(2)\end{array}$

When $x=0$

  $\begin{array}{l}{\left(0\right)}^2-4y^2=4\\ -4y^2=4\\ y^2=-1\end{array}$  And  $\begin{array}{l}6y^2-\left(0\right)=2\\ y^2=\frac{1}{3}\\ y=\pm \frac{1}{\sqrt{3}}\end{array}$

No solution

When $y=0$

  $\begin{array}{l}{x}^2-4{\left(0\right)}^2=4\\ x^2=4\\ x^2=\pm 2\end{array}$  And  $\begin{array}{l}6{\left(0\right)}^2-x=2\\ x=-2\end{array}$

Plotting the graph of equations on same plane:

  

From graph, the two equations intersects at three points

Conclusion:

Hence, number of real solutions is $3$