Question

solve the following equation for x.
$\frac{10}{9}-(5x - 2)=\frac{1}{4}x-12$
a. $\frac{7}{2}$
b. $\frac{2}{7}$
c. $\frac{27}{7}$
d. $-\frac{7}{2}$

Explanation:

Step1: Expand the left - hand side

\[-\frac{9}{10}(5x - 2)=-\frac{9}{10}\times5x+\frac{9}{10}\times2=-\frac{9}{2}x+\frac{9}{5}\]
The original equation \(-\frac{9}{10}(5x - 2)=\frac{1}{4}x-12\) becomes \(-\frac{9}{2}x+\frac{9}{5}=\frac{1}{4}x - 12\).

Step2: Move the \(x\) terms to one side

Add \(\frac{9}{2}x\) to both sides:
\(\frac{9}{5}=\frac{1}{4}x+\frac{9}{2}x - 12\).
Find a common denominator for the \(x\) terms on the right - hand side. The common denominator of 4 and 2 is 4, so \(\frac{1}{4}x+\frac{9}{2}x=\frac{1}{4}x+\frac{18}{4}x=\frac{1 + 18}{4}x=\frac{19}{4}x\).
The equation is now \(\frac{9}{5}=\frac{19}{4}x-12\).

Step3: Move the constant terms to one side

Add 12 to both sides. Since \(12=\frac{60}{5}\), we have \(\frac{9}{5}+\frac{60}{5}=\frac{19}{4}x\), so \(\frac{9 + 60}{5}=\frac{19}{4}x\), which simplifies to \(\frac{69}{5}=\frac{19}{4}x\).

Step4: Solve for \(x\)

Multiply both sides by \(\frac{4}{19}\):
\(x=\frac{69}{5}\times\frac{4}{19}=\frac{276}{95}\)

Answer:

\(x = \frac{276}{95}\)