Question

Sketch the graph if 2x + 3y - z = 6 and state it's x-, y-, and z-intercepts.

Answer 1

Problem Statement

The question asks us to plot the function:

`2x+3y-z=6`

And we were asked to state the x-, y- and z- intercepts.

Solution

The equation given is a 3-d equation as such when it is plotted, it gives a plane instead of a line for a normal linear 2-D equation.

We shall begin with the intercepts. To find the intercepts, we simply set x, y to zero to find the z-intercept, x, z to zero to find the y-intercept, and y, z to zero to find the x-intercept.

Let us proceed to find the intercepts:

`\begin{gathered} 2x+3y-z=6 \\ X-\text{intercept: set y and z to zero} \\ 2x+3(0)-0=6 \\ 2x=6 \\ \text{divide both sides by 2} \\ x=3 \\ \\ Y-\text{intercept: set x and z to zero} \\ 2(0)+3y-0=6 \\ 3y=6 \\ \text{Divide both sides by 3} \\ y=2 \\ \\ Z-\text{intercept: set x and y to zero:} \\ 2(0)+3(0)-z=6 \\ \therefore z=-6 \end{gathered}`

The intercepts are:

x-intercept = 3

y-intercept = 2

z-intercept = -6

Graphically, we can observe these intercepts shown below:

The above figure shows the x and z intercepts.

The y-intercept is shown below:

The final image of the plane will look like this: