Question

Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form.

$\frac{2}{3}x-\frac{1}{3}y=2$

Short answer

Yes, the given equation$\frac{2}{3}x-\frac{1}{3}y=2$ is a linear equation.

The given equation$\frac{2}{3}x-\frac{1}{3}y=2$ in standard form is $2x-y=6$.

Step-by-step solution

Step 1. Determine whether the given equation is a linear equation.

A linear equation is an equation that forms a line when it is graphed. Linear equations are written in the form $Ax+By=C$, where $A$and $B$are not both zero and localid="1647343581190" $A$,$B$and$C$are integers with greatest common factor of 1.

The given equation is $\frac{2}{3}x-\frac{1}{3}y=2$.

The given equation can be written as:

$$\begin{gathered} \frac{2}{3}x-\frac{1}{3}y=2 \\ \frac{2x}{3}-\frac{y}{3}=2 \\ \frac{2x-y}{3}=2 \\ 2x-y=6 \end{gathered}$$

Therefore, by comparing the given equation$2x-y=6$with the equation $Ax+By=C$, it can be noticed that:

$A=2$,$B=-1$and $C=6$.

As, the equation$\frac{2}{3}x-\frac{1}{3}y=2$can be written in the form $Ax+By=C$.

Therefore, yes, the given equation$\frac{2}{3}x-\frac{1}{3}y=2$is a linear equation.

Step 2. Write the given equation in standard form.

The given equation is $\frac{2}{3}x-\frac{1}{3}y=2$.

The standard form of the linear equation is $Ax+By=C$.

The given equation in standard form is:

$2x-y=6$

Where $A=2$,$B=-1$ and $C=6$.