Question

solve the following equation for x. (\frac{9}{10}-(5x - 2)=\frac{1}{4}x-12). a. (\frac{102}{17}) b. (-\frac{544}{171}) c. (\frac{544}{189}) d. (-\frac{320}{189})

Explanation:

Step1: Expand the left - hand side

First, expand \(\frac{10}{9}-(5x - 2)\) to get \(\frac{10}{9}-5x + 2=\frac{10}{9}+2-5x=\frac{10 + 18}{9}-5x=\frac{28}{9}-5x\). The original equation \(\frac{10}{9}-(5x - 2)=\frac{1}{4}x-12\) becomes \(\frac{28}{9}-5x=\frac{1}{4}x-12\).

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

Add \(5x\) to both sides of the equation: \(\frac{28}{9}=\frac{1}{4}x + 5x-12\). Combine the \(x\) terms: \(\frac{1}{4}x+5x=\frac{1 + 20}{4}x=\frac{21}{4}x\). So the equation is \(\frac{28}{9}=\frac{21}{4}x-12\).

Step3: Move the constant terms to one side

Add 12 to both sides: \(\frac{28}{9}+12=\frac{21}{4}x\). Calculate \(\frac{28}{9}+12=\frac{28+108}{9}=\frac{136}{9}\). So \(\frac{136}{9}=\frac{21}{4}x\).

Step4: Solve for \(x\)

Multiply both sides by \(\frac{4}{21}\) to isolate \(x\): \(x=\frac{136}{9}\times\frac{4}{21}=\frac{544}{189}\).

Answer:

C. \(\frac{544}{189}\)