Chapter 1.8, Problem 55E

To determine

To calculate: The x- and y- intercepts for the equation $x^4+y^2-xy=16$ and the graph of the equation is provided below,

  

Answer to Problem 55E

The x-intercepts are $-2$ and $2$andy-intercepts are $-4$ and $4$ .

Explanation of Solution

Given information:

The equation $x^4+y^2-xy=16$ and the graph of the equation is provided below,

  

Formula used:

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Calculation:

Consider the provided equation $x^4+y^2-xy=16$ and the graph of the equation is sketched below,

  

Observe from the graph that the curve intersects x-axis at two points $-2$ and $2$ also y-axis at two points $-4$ and $4$ .

Recall that the x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

Substitute $y=0$ in the equation $x^4+y^2-xy=16$ ,

  $\begin{array}{c}{x}^4+0^2-x\left(0\right)=16\\ x^4=16\\ x=\pm 2\end{array}$

Therefore, x-intercepts are $-2$ and 2.

Recall that the y-intercepts are the points on x-axis where the graph of the equation intersects the y-axis.

Substitute $x=0$ in the equation $x^4+y^2-xy=16$ ,

  $\begin{array}{c}{0}^4+y^2-\left(0\right)y=16\\ y^2=16\\ y=\pm 4\end{array}$

Therefore, y-intercepts are $-4$ and $4$ .

Thus, the x-intercepts are $-2$ and $2$andy-intercepts are $-4$ and $4$ .