Question

Rahul is presented with the equation of a line in slope-intercept form: y= 2/3x + 1/4. By what number can he multiply the equation in order to clear the fractions and rewrite the equation in standard form?

Answer 1

To clear the fractions from the equation \( y = \frac{2}{3}x + \frac{1}{4} \), we need to find a common denominator for the fractions. The denominators present in the equation are 3 and 4.

The least common multiple (LCM) of 3 and 4 is 12. Thus, Rahul can multiply the entire equation by 12 to eliminate the fractions.

Let's perform the multiplication:

\[
12y = 12\left(\frac{2}{3}x\right) + 12\left(\frac{1}{4}\right)
\]

Calculating each term:

1. For \( 12\left(\frac{2}{3}x\right) \):
\[
12 \cdot \frac{2}{3} = 4 \cdot 2 = 8
\]
So this term becomes \( 8x \).

2. For \( 12\left(\frac{1}{4}\right) \):
\[
12 \cdot \frac{1}{4} = 3
\]
So this term becomes \( 3 \).

Thus, we rewrite the equation as:

\[
12y = 8x + 3
\]

To have the equation in standard form (Ax + By = C), we can rearrange it as follows:

\[
-8x + 12y = 3
\]

Now, if we multiply the entire equation by -1 for a standard positive coefficient on \( x \):

\[
8x - 12y = -3
\]

In conclusion, Rahul should multiply the entire equation by **12** to clear the fractions and write the equation in standard form.