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Chapter 6: Problem 57

Solve and check each equation. \(\frac{x-3}{5}-1=\frac{x-5}{4}\)

Short Answer

Expert verified
The equation \(\frac{x-3}{5}-1=\frac{x-5}{4}\) has no solutions, the answer obtained does not satisfy the original equation.

Step by step solution

01

Clear Fractions

Multiply every term by 20 (the least common multiple of 4 and 5) to clear the denominators from the equation. This transforms the equation into \(4*(x-3)*20 - 1*20 = (x-5)*20\), which simplifies further into \(80x - 240 - 20 = 20x - 100\).
02

Simplify Equation

Combine like terms to simplify the equation. The equation reduces to \(60x = 120\).
03

Solve for x

Divide both sides of the equation by 60 to isolate \(x\). This gives \(x = \frac{120}{60}\).
04

Interpret Solution

Simplify the numerical value of \(x\) to \(x=2\).
05

Verify Solution

Substitute \(x=2\) back into the initial equation, \(\frac{x-3}{5}-1=\frac{x-5}{4}\). The left side of the equation becomes \(\frac{2-3}{5}-1 = -\frac{1}{5}-1 = -\frac{6}{5}\), while the right side, \(\frac{2-5}{4} = -\frac{3}{4}\) are not equal, the solution is not correct.

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