Q4. - Algebra 1

Question

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

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

Step-by-Step Solution

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 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 $A$, $B$ 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 $C=6$.