Chapter 9, Problem 12CT

To determine

To find the number of real solutions of each system - $4x^2-y^2=4$ and $xy=-9$

Answer to Problem 12CT

There are two real solutions to the system $4x^2-y^2=4$ and no solution to the system $xy=-9$

Explanation of Solution

Given:

Equations: $4x^2-y^2=4$ and $xy=-9$

Formula Used:

Real solution of a system is defined when the equation intersects the x-axis.

Calculation:

Given:

Equations: $4x^2-y^2=4$ and $xy=-9$

Plotting $4x^2-y^2=4$ on a graph, we have:

  

From the above graph, we can see that the equation intersects x-axis at two points.

Thus, there are two real solutions to the above system and those are $\left(-1,0\right)$ and $\left(1,0\right)$ .

Plotting $xy=-9$ on a graph, we have:

  

From the above graph, we can see that the equation does not intersect x-axis at any points

Thus, there are no real solutions to the above system.

Conclusion:

There are two real solutions to the system $4x^2-y^2=4$ and no solution to the system $xy=-9$