Chapter P, Problem 9T

To determine

To calculate: The intercepts of the equation $y=4-\frac{3}{4}x$ and test for symmetry. Then find the graph of the equation.

The intercepts of the equation $y=4-\frac{3}{4}x$ are $\left(\frac{16}{3},0\right)$ and $\left(0,4\right)$.

The equation $y=4-\frac{3}{4}x$ is neither symmetric about $x\text{-axis}$ or $y\text{-axis}$, nor about the origin.

Calculation:

Consider the provided equation $y=4-\frac{3}{4}x$.

To find the $x\text{-intercept}$ of the equation $y=4-\frac{3}{4}x$ substitute $0$ for $y$,

$0=4-\frac{3}{4}x$

Simplify the equation to get the value of $x$,

$\begin{array}{c}4-\frac{3}{4}x=0\\ 4=\frac{3}{4}x\text{ Add }\frac{3}{4}x\text{ on the both sides}\\ 16=3x\text{ Multiply by 4 to each side}\\ x=\frac{16}{3}\text{ Divide by 3 to each side}\end{array}$

Hence, the $x\text{-intercept}$ of the equation $y=4-\frac{3}{4}x$ is $\left(\frac{16}{3},0\right)$.

To find the $y\text{-intercept}$ of the equation $y=4-\frac{3}{4}x$ substitute $0$ for $x$,

$y=4-\frac{3}{4}\left(0\right)$

Simplify the equation to get the value of $y$.

$y=4$

Hence, the $y\text{-intercept}$ of the equation $y=4-\frac{3}{4}x$ is $\left(0,4\right)$.

To check the symmetry about origin, replace $x$ by $-x$ and $y$ by $-y$ in the provided equation $y=4-\frac{3}{4}x$,

$\begin{array}{c}-y=4-\frac{3}{4}\left(-x\right)\\ y=-4+\frac{3}{4}x\end{array}$

Which is not equal to the equivalent equation. Hence, the given equation is not symmetric about the origin.

To check the symmetry about $x\text{-axis}$ replace $y$ by $-y$ in the provided equation $y=4-\frac{3}{4}x$,

$\begin{array}{r}-y=4-\frac{3}{4}\left(x\right)\\ y=-4+\frac{3}{4}x\end{array}$

which is not equal to the equivalent equation. Hence, the given equation is not symmetric about $x\text{-axis}$.

To check the symmetry about $y\text{-axis}$ replace $x$ by $-x$ in the provided equation $y=4-\frac{3}{4}x$,

$\begin{array}{c}y=4-\frac{3}{4}\left(-x\right)\\ y=4+\frac{3}{4}x\end{array}$

Which is not equal to the equivalent equation. Hence, the given equation is not symmetric about $y\text{-axis}$.

Therefore the equation $y=4-\frac{3}{4}x$ is neither symmetric about $x-\text{axis}$ or $y-\text{axis}$ nor about the origin.

Graph:

To draw the graph of the equation $y=4-\frac{3}{4}x$, make a table of coordinates,

$\begin{array}{ccc}x& y=4-\frac{3}{4}x& \left(x,y\right)\\ 0& 4& \left(0,4\right)\\ \frac{16}{3}& 0& \left(\frac{16}{3},0\right)\end{array}$

Thus, the graph is shown below,

Hence, the intercepts of the equation $y=4-\frac{3}{4}x$ are $\left(\frac{16}{3},0\right)$ and $\left(0,4\right)$ and the equation $y=4-\frac{3}{4}x$ is neither symmetric about $x\text{-axis}$ or $y\text{-axis}$, nor about the origin.