Chapter 7.1, Problem 66E

Solution

a.

To calculate: The solution of equation $\frac{x^2}{4}-\frac{y^2}{9}=1$ in the terms for y in terms of x to obtained two function.

The obtained matrix is $y=\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)},y=-\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}$ .

Given Information:

The equation is $\frac{x^2}{4}-\frac{y^2}{9}=1$ .

Calculation:

Consider the given equation,

  $\frac{x^2}{4}-\frac{y^2}{9}=1$

Now, solve the equation.

  $\begin{array}{c}\frac{x^2}{4}-\frac{y^2}{9}=1\\ \frac{y^2}{9}=\frac{x^2}{4}-1\\ y^2=\frac{1}{9}\left(\frac{x^2}{4}-1\right)\\ y=\pm \frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}\end{array}$

So, the functions are defined as,

  $y=\frac{1}{3}\sqrt{\left(1-\frac{x^2}{4}\right)},y=-\frac{1}{3}\sqrt{\left(1-\frac{x^2}{4}\right)}$

Therefore, the obtained functions is $y=\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)},y=-\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}$ .

b.

To calculate: The solution of system of the equation by using graph.

The obtained result is $\left(-2.683,-2.683\right),\left(2.683,2.683\right)$ .

Given Information:

The obtained system of equation is $y=\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)},y=-\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}$ and $x-y=0$ .

Calculation:

Consider the given information,

  $y=\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)},y=-\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}$ and $x-y=0$ .

Use the graphing calculator to graph system and find the solution.

  

Hence, the obtained result is $\left(-2.683,-2.683\right),\left(2.683,2.683\right)$ .

c.

To calculate: The solution of system of the equation by substitution method.

The obtained result is $\left(-2.683,-2.683\right),\left(2.683,2.683\right)$ .

Given Information:

The obtained system of equation is $y=\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)},y=-\frac{1}{3}\sqrt{\left(\frac{x^2}{4}-1\right)}$ and $x-y=0$ .

Calculation:

Consider the given information,

  $\frac{x^2}{4}-\frac{y^2}{9}=1$ and $x-y=0$ .

Substitute $y=x$ in the equation $\frac{x^2}{4}-\frac{y^2}{9}=1$ .

  $\begin{array}{c}\frac{x^2}{4}-\frac{x^2}{9}=1\\ \frac{9x^2-4x^2}{36}=1\\ x^2=\frac{36}{5}\\ x=\pm \text{2}\text{.683}\end{array}$

Hence, the obtained result is $\left(-2.683,-2.683\right),\left(2.683,2.683\right)$ .