Question

question 16
solve the following.
\\(\frac{1}{2}x + \frac{1}{4} = -\frac{3}{4}x + \frac{1}{2}\\)
\\(x = \\)
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Explanation:

Step1: Add $\frac{3}{4}x$ to both sides

To get all the $x$ terms on one side, we add $\frac{3}{4}x$ to both sides of the equation $\frac{1}{2}x+\frac{1}{4}=-\frac{3}{4}x+\frac{1}{2}$.
\[
\frac{1}{2}x+\frac{3}{4}x+\frac{1}{4}=-\frac{3}{4}x+\frac{3}{4}x+\frac{1}{2}
\]
Simplifying the left side: $\frac{1}{2}x+\frac{3}{4}x=\frac{2}{4}x+\frac{3}{4}x=\frac{5}{4}x$, and the right side: $-\frac{3}{4}x+\frac{3}{4}x = 0$, so we have $\frac{5}{4}x+\frac{1}{4}=\frac{1}{2}$.

Step2: Subtract $\frac{1}{4}$ from both sides

To isolate the term with $x$, we subtract $\frac{1}{4}$ from both sides.
\[
\frac{5}{4}x+\frac{1}{4}-\frac{1}{4}=\frac{1}{2}-\frac{1}{4}
\]
Simplifying the left side: $\frac{1}{4}-\frac{1}{4}=0$, so we have $\frac{5}{4}x=\frac{1}{2}-\frac{1}{4}$.
Calculating the right side: $\frac{1}{2}-\frac{1}{4}=\frac{2}{4}-\frac{1}{4}=\frac{1}{4}$. So now the equation is $\frac{5}{4}x=\frac{1}{4}$.

Step3: Multiply both sides by $\frac{4}{5}$

To solve for $x$, we multiply both sides by the reciprocal of $\frac{5}{4}$, which is $\frac{4}{5}$.
\[
x=\frac{1}{4}\times\frac{4}{5}
\]
Simplifying the right side: $\frac{1}{4}\times\frac{4}{5}=\frac{1\times4}{4\times5}=\frac{1}{5}$.

Answer:

$\frac{1}{5}$