Question

Rewrite the following equation in standard form.

y
−
5
3
=
3
4
(
x
+ 1
)

Answer 1

To rewrite the equation \( y - \frac{5}{3} = \frac{3}{4}(x + 1) \) in standard form, we first eliminate the fractions and rearrange the equation.

1. Start with the original equation:
\[
y - \frac{5}{3} = \frac{3}{4}(x + 1)
\]

2. Distribute \(\frac{3}{4}\):
\[
y - \frac{5}{3} = \frac{3}{4}x + \frac{3}{4}
\]

3. Add \(\frac{5}{3}\) to both sides:
\[
y = \frac{3}{4}x + \frac{3}{4} + \frac{5}{3}
\]

4. To combine the constants on the right side, we need a common denominator for \(\frac{3}{4}\) and \(\frac{5}{3}\). The least common multiple of 4 and 3 is 12. Rewrite each fraction:
\[
\frac{3}{4} = \frac{9}{12}, \quad \frac{5}{3} = \frac{20}{12}
\]

5. Now combine the fractions:
\[
y = \frac{3}{4}x + \left(\frac{9}{12} + \frac{20}{12}\right) = \frac{3}{4}x + \frac{29}{12}
\]

6. To eliminate the fraction from the equation, multiply every term by 12:
\[
12y = 12 \cdot \frac{3}{4}x + 29
\]
This simplifies to:
\[
12y = 9x + 29
\]

7. Rearrange to get the equation in standard form \(Ax + By = C\):
\[
-9x + 12y = 29
\]

This can also be expressed by multiplying through by -1 to write it with the \(x\) term positive:
\[
9x - 12y = -29
\]

Thus, the equation in standard form is:
\[
9x - 12y = -29
\]