Question

question 3 of 5
select the correct answer.
what is the solution for w in the equation?
w + \frac{1}{2} = \frac{1}{4}w + 2
\bigcirc \\ w = \frac{9}{8}
\bigcirc \\ w = 2
\bigcirc \\ w = 4
\bigcirc \\ w = \frac{10}{3}

Explanation:

Step1: Subtract $\frac{1}{4}w$ from both sides

To get all the \( w \)-terms on one side, we subtract \( \frac{1}{4}w \) from both sides of the equation \( w+\frac{1}{2}=\frac{1}{4}w + 2 \).
\( w-\frac{1}{4}w+\frac{1}{2}=\frac{1}{4}w-\frac{1}{4}w + 2 \)
Simplifying the left side: \( w-\frac{1}{4}w=\frac{4}{4}w-\frac{1}{4}w=\frac{3}{4}w \), and the right side becomes \( 2 \). So the equation is now \( \frac{3}{4}w+\frac{1}{2}=2 \).

Step2: Subtract $\frac{1}{2}$ from both sides

To isolate the term with \( w \), we subtract \( \frac{1}{2} \) from both sides.
\( \frac{3}{4}w+\frac{1}{2}-\frac{1}{2}=2-\frac{1}{2} \)
Simplifying the left side, the \( \frac{1}{2} \) terms cancel out, and the right side: \( 2-\frac{1}{2}=\frac{4}{2}-\frac{1}{2}=\frac{3}{2} \). So now we have \( \frac{3}{4}w=\frac{3}{2} \).

Step3: Solve for $w$

To solve for \( w \), we multiply both sides by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \).
\( w=\frac{3}{2}\times\frac{4}{3} \)
Simplifying the right side: the 3s cancel out, and \( \frac{4}{2} = 2 \)? Wait, no, wait: \( \frac{3}{2}\times\frac{4}{3}=\frac{3\times4}{2\times3}=\frac{4}{2}=2 \)? Wait, no, wait, let's recalculate. Wait, \( \frac{3}{2}\times\frac{4}{3}=\frac{4}{2}=2 \)? Wait, no, that's not right. Wait, no, wait, in step 2, \( 2-\frac{1}{2}=\frac{3}{2} \)? Wait, \( 2=\frac{4}{2} \), so \( \frac{4}{2}-\frac{1}{2}=\frac{3}{2} \), correct. Then \( \frac{3}{4}w=\frac{3}{2} \). Multiply both sides by \( \frac{4}{3} \): \( w=\frac{3}{2}\times\frac{4}{3}=\frac{12}{6}=2 \)? Wait, no, \( \frac{3\times4}{2\times3}=\frac{4}{2}=2 \). Wait, but let's check the original equation. Let's plug \( w = 2 \) into the original equation: left side: \( 2+\frac{1}{2}=\frac{5}{2} \), right side: \( \frac{1}{4}\times2 + 2=\frac{1}{2}+2=\frac{5}{2} \). So that works. Wait, but let's check the other options. Wait, maybe I made a mistake. Wait, no, let's re - do the steps.

Wait, original equation: \( w+\frac{1}{2}=\frac{1}{4}w + 2 \)

Subtract \( \frac{1}{4}w \) from both sides: \( w-\frac{1}{4}w+\frac{1}{2}=2 \)

\( \frac{3}{4}w+\frac{1}{2}=2 \)

Subtract \( \frac{1}{2} \) from both sides: \( \frac{3}{4}w=2 - \frac{1}{2}=\frac{3}{2} \)

Multiply both sides by \( \frac{4}{3} \): \( w=\frac{3}{2}\times\frac{4}{3}=\frac{12}{6}=2 \). So \( w = 2 \) is the solution.

Wait, but let's check the option \( w=\frac{10}{3} \). Wait, maybe I made a mistake. Wait, no, let's re - solve:

\( w+\frac{1}{2}=\frac{1}{4}w + 2 \)

Subtract \( \frac{1}{4}w \): \( \frac{3}{4}w+\frac{1}{2}=2 \)

Subtract \( \frac{1}{2} \): \( \frac{3}{4}w=\frac{3}{2} \)

Multiply by \( \frac{4}{3} \): \( w=\frac{3}{2}\times\frac{4}{3}=2 \). Yes, that's correct. So the solution is \( w = 2 \).

Answer:

\( w = 2 \) (corresponding to the option: \( w = 2 \))