Question

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

$\frac{1}{4}x-\frac{3}{4}y=-1$

Short answer

Yes,equation is a linear equation and the standard form is $4x-12y+4=0$.

Step-by-step solution

Step 1. State the concept of standard form of linear equation.

Alinear equationis anyequationthat can be written in the form$ax+by=0$, where a and b are real numbers and x is a variable. This form is sometimes called the standard form of a linear equation.

No variable in alinear equationis raised to a power greater than 1. Linear equationsare those equations that are of the first order.

Standard form of Linear Equation is$Ax+By+C=0$, where A, B and C are any integer.

Step 2. Rewrite the equation in standard form.

In order to rewrite the equation $\frac{1}{4}x-\frac{3}{4}y=-1$ in standard form multiply each term of the equation by 4.

$$\begin{gathered} \frac{1}{4}x-\frac{3}{4}y=-1 \\ 4\cdot \frac{1}{4}x-4\cdot \frac{3}{4}y=-1\cdot 4 \\ 4x-12y=-4 \\ 4x-12y+4=-4+4\text{\hspace{0.17em}[\hspace{0.17em}Add\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}both\hspace{0.17em}\hspace{0.17em}sides\hspace{0.17em}]} \\ 4x-12y+4=0 \end{gathered}$$

Step 3. State the Conclusion.

The equation is now in standard form where$A=4,B=-12\text{and}C=4$. Since the variables in the equation has the highest exponent as 1, so the equation is linear.

Yes, this is a linear equation and the standard form is:$4x-12y+4=0$.