Question

(\frac{x - 1}{2}=12) (\frac{2}{4}=\frac{6}{-2x + 8}) (2(10 - 3x)=6x + 5 - 12x) (\frac{1}{2}(18x + 100)=20 - x)

Explanation:

1. Solve \(\boldsymbol{\frac{x - 1}{2}=12}\)

Step1: Eliminate the denominator

Multiply both sides of the equation by 2 to get rid of the denominator on the left - hand side.
\(2\times\frac{x - 1}{2}=12\times2\)
Simplifying the left - hand side, we have \(x - 1 = 24\).

Step2: Solve for \(x\)

Add 1 to both sides of the equation \(x-1 = 24\).
\(x-1 + 1=24 + 1\)
So, \(x = 25\).

Step1: Cross - multiply

Cross - multiply the proportion \(\frac{2}{4}=\frac{6}{-2x + 8}\), we get \(2\times(-2x + 8)=4\times6\).

Step2: Expand and simplify

Expand the left - hand side: \(2\times(-2x)+2\times8 = 24\), which is \(-4x + 16 = 24\).
Subtract 16 from both sides: \(-4x+16 - 16=24 - 16\), so \(-4x = 8\).
Divide both sides by - 4: \(\frac{-4x}{-4}=\frac{8}{-4}\), and \(x=-2\).

Step1: Expand the left - hand side and simplify the right - hand side

Expand the left - hand side: \(2\times10-2\times3x=20 - 6x\).
Simplify the right - hand side: \(6x-12x + 5=-6x + 5\).
The equation becomes \(20 - 6x=-6x + 5\).

Step2: Analyze the equation

Add \(6x\) to both sides of the equation: \(20-6x + 6x=-6x + 5+6x\).
We get \(20 = 5\), which is a contradiction. So, this equation has no solution.

Answer:

\(x = 25\)

2. Solve \(\boldsymbol{\frac{2}{4}=\frac{6}{-2x + 8}}\)