Question

linear equations
solving a two - step equation with signed fractions
solve for y.
$-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}$
simplify your answer as much as possible.
$y = \square$

Explanation:

Step1: Add $\frac{1}{4}$ to both sides

To isolate the term with \( y \), we add \(\frac{1}{4}\) to both sides of the equation \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\).
\[ -\frac{9}{8}y - \frac{1}{4}+\frac{1}{4}=-\frac{3}{2}+\frac{1}{4} \]
Simplifying the left side, \(-\frac{1}{4}+\frac{1}{4} = 0\), so we have \(-\frac{9}{8}y=-\frac{3}{2}+\frac{1}{4}\).
For the right side, find a common denominator (which is 4) and rewrite \(-\frac{3}{2}\) as \(-\frac{6}{4}\). Then \(-\frac{6}{4}+\frac{1}{4}=-\frac{5}{4}\)? Wait, no, wait: \(-\frac{3}{2}=-\frac{6}{4}\), so \(-\frac{6}{4}+\frac{1}{4}=-\frac{5}{4}\)? Wait, no, let's recalculate: \(-\frac{3}{2}+\frac{1}{4}\). The common denominator is 4. \(-\frac{3}{2}=-\frac{6}{4}\), so \(-\frac{6}{4}+\frac{1}{4}=\frac{-6 + 1}{4}=\frac{-5}{4}\)? Wait, no, the original equation is \(-\frac{3}{2}\), so when we add \(\frac{1}{4}\), it's \(-\frac{3}{2}+\frac{1}{4}\). Let's do it correctly: \(-\frac{3}{2}=-\frac{6}{4}\), so \(-\frac{6}{4}+\frac{1}{4}=\frac{-6 + 1}{4}=\frac{-5}{4}\)? Wait, but that seems wrong. Wait, the original equation is \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\). So adding \(\frac{1}{4}\) to both sides: \(-\frac{9}{8}y=-\frac{3}{2}+\frac{1}{4}\). Let's compute \(-\frac{3}{2}+\frac{1}{4}\). Convert \(-\frac{3}{2}\) to fourths: \(-\frac{6}{4}\). Then \(-\frac{6}{4}+\frac{1}{4}=\frac{-6 + 1}{4}=\frac{-5}{4}\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the right side is \(-\frac{3}{2}\), and we add \(\frac{1}{4}\). So \(-\frac{3}{2}+\frac{1}{4}\). Let's calculate: \(-\frac{3}{2}= -1.5\), \(\frac{1}{4}=0.25\), so \(-1.5 + 0.25=-1.25\), which is \(-\frac{5}{4}\)? Wait, no, \(-1.25\) is \(-\frac{5}{4}\)? Wait, \(\frac{5}{4}=1.25\), so \(-\frac{5}{4}=-1.25\). But let's check again. Wait, maybe I messed up the sign. Wait, the original equation is \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\). So we add \(\frac{1}{4}\) to both sides: \(-\frac{9}{8}y=-\frac{3}{2}+\frac{1}{4}\). Let's compute \(-\frac{3}{2}+\frac{1}{4}\). \(-\frac{3}{2}\) is \(-1.5\), \(\frac{1}{4}\) is \(0.25\), so \(-1.5 + 0.25=-1.25\), which is \(-\frac{5}{4}\)? Wait, no, \(-1.25\) is \(-\frac{5}{4}\)? Wait, \(\frac{5}{4}=1.25\), so \(-\frac{5}{4}=-1.25\). So \(-\frac{9}{8}y=-\frac{5}{4}\)? Wait, no, wait, maybe I made a mistake in the sign. Wait, the original equation is \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\). So if we add \(\frac{1}{4}\) to both sides, we get \(-\frac{9}{8}y=-\frac{3}{2}+\frac{1}{4}\). Let's compute \(-\frac{3}{2}+\frac{1}{4}\) again. \(-\frac{3}{2}\) is equal to \(-\frac{6}{4}\), so \(-\frac{6}{4}+\frac{1}{4}=\frac{-6 + 1}{4}=\frac{-5}{4}\). So \(-\frac{9}{8}y=-\frac{5}{4}\)? Wait, no, that doesn't seem right. Wait, maybe I flipped the sign. Wait, let's start over.

Original equation: \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\)

Step 1: Add \(\frac{1}{4}\) to both sides:

Left side: \(-\frac{9}{8}y - \frac{1}{4}+\frac{1}{4}=-\frac{9}{8}y\)

Right side: \(-\frac{3}{2}+\frac{1}{4}\)

Convert \(-\frac{3}{2}\) to fourths: \(-\frac{6}{4}\)

So \(-\frac{6}{4}+\frac{1}{4}=\frac{-6 + 1}{4}=\frac{-5}{4}\)? Wait, no, \(-6 + 1 = -5\), so \(\frac{-5}{4}\). So \(-\frac{9}{8}y=-\frac{5}{4}\)? Wait, but let's check with another approach. Let's multiply both sides by 8 to eliminate denominators.

Original equation: \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\)

Multiply every term by 8:

\(8\times(-\frac{9}{8}y)-8\times\frac{1}{4}=8\times(-\frac{3}{2})\)

Simplify each term:

\(-9y - 2 = -12\)

Ah, that's a better approach. So Step 1: Multiply all terms by 8 to eliminate fractions.

\(8\times(-\frac{9}{8}y)= -9y\)

\(8\times(-\frac{1}{4})= -2\)

\(8\times(-\frac{3}{2})= -12\)

So the equation becomes \(-9y - 2 = -12\)

Step 2: Add 2 to both sides

Add 2 to both sides to isolate the term with \( y \):

\(-9y - 2 + 2 = -12 + 2\)

Simplify: \(-9y = -10\)

Step 3: Divide both sides by -9

Divide both sides by -9 to solve for \( y \):

\(y=\frac{-10}{-9}=\frac{10}{9}\)

Wait, let's check with the first method. If we have \(-\frac{9}{8}y=-\frac{5}{4}\), then multiply both sides by \(-\frac{8}{9}\):

\(y=(-\frac{5}{4})\times(-\frac{8}{9})=\frac{40}{36}=\frac{10}{9}\). Yes, that matches. So my initial mistake was in the first step's arithmetic, but using the common denominator or multiplying by 8 (the least common multiple of 8, 4, 2) is better.

So let's redo Step 1 correctly by multiplying by 8:

Original equation: \(-\frac{9}{8}y - \frac{1}{4}=-\frac{3}{2}\)

Multiply each term by 8:

\(8\times(-\frac{9}{8}y) = -9y\)

\(8\times(-\frac{1}{4}) = -2\)

\(8\times(-\frac{3}{2}) = -12\)

So equation: \(-9y - 2 = -12\)

Step 2: Add 2 to both sides:

\(-9y - 2 + 2 = -12 + 2\)

\(-9y = -10\)

Step 3: Divide by -9:

\(y = \frac{-10}{-9} = \frac{10}{9}\)

Let's verify the solution. Substitute \( y = \frac{10}{9} \) into the original equation:

Left side: \(-\frac{9}{8}\times\frac{10}{9}-\frac{1}{4}\)

Simplify \(-\frac{9}{8}\times\frac{10}{9}\): the 9s cancel, so \(-\frac{10}{8}=-\frac{5}{4}\)

Then \(-\frac{5}{4}-\frac{1}{4}=-\frac{6}{4}=-\frac{3}{2}\), which matches the right side. So the solution is correct.

Answer:

\(y = \frac{10}{9}\)