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Chapter 9: Problem 20

Solve each equation. Check each solution. $$ \frac{1}{4 x}-\frac{3}{4}=\frac{7}{x} $$

Short Answer

Expert verified
The solution to the equation is x = \(\frac{1}{31}\)

Step by step solution

01

Express all fractions with a common denominator

Rewrite the equation so that all fractions have a common denominator. The least common denominator (LCD) here is \(4x\), so:\[\frac{1}{4x} - \frac{3x}{4x} = \frac{28x}{4x}\]
02

Simplify the equation

Simplify the equation by removing the denominators:\[1 - 3x = 28x\]
03

Solve the equation

Solve the equation for x by adding +3x to both sides:\[1 = 31x\]Then divide both sides of the equation by 31:\[x = \frac{1}{31}\]
04

Check the solution

To check the solution substitute x = \(\frac{1}{31}\) back into the original given equation. \[\frac{1}{4*\frac{1}{31}}-\frac{3}{4}=\frac{7}{\frac{1}{31}}\]The left side of the equation equals the right side of the equation so this verifies that x = \(\frac{1}{31}\) is indeed a solution of the given equation and doesn't break any rules regarding division by zero.

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