Chapter 8.7, Problem 36E

Solution

To calculate: The solution of given system of equations by substitution.

The given system has infinite number of solutions.

Given information: The given system of equations is:

  $\begin{array}{l}\frac{x^2}{2}+\frac{y^2}{4}=1\\ 4y^2=16-8x^2\end{array}$

Calculation:

Consider the first equation.

  $\frac{x^2}{2}+\frac{y^2}{4}=1$ …… (1)

Consider the second equation.

  $4y^2=16-8x^2$ …… (2)

Simplify the first equation.

  $\begin{array}{c}\frac{2x^2+y^2}{4}=1\\ 2x^2+y^2=4\end{array}$

Multiply the above equation by $4$.

  $\begin{array}{c}4\left(2x^2+y^2\right)=4\left(4\right)\\ 8x^2+4y^2=16\\ 4y^2=16-8x^2\end{array}$

It can be observed that the above equation is the same as the equation (2). So, the system has infinite number of solutions.

Therefore, the given system has infinite number of solutions.