Question

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

Explanation:

First Equation: $\boldsymbol{\frac{2}{4}=\frac{6}{-2x + 8}}$

Step1: Cross - multiply

Cross - multiplying the proportion $\frac{a}{b}=\frac{c}{d}$ gives $a\times d=b\times c$. So for $\frac{2}{4}=\frac{6}{-2x + 8}$, we have $2\times(-2x + 8)=4\times6$.

Step2: Simplify both sides

Simplify the left - hand side: $2\times(-2x)+2\times8=-4x + 16$.
Simplify the right - hand side: $4\times6 = 24$.
So the equation becomes $-4x+16 = 24$.

Step3: Solve for x

Subtract 16 from both sides: $-4x+16 - 16=24 - 16$.
We get $-4x=8$.
Divide both sides by - 4: $x=\frac{8}{-4}=-2$.

Second Equation: $\boldsymbol{2(10 - 3x)=6x + 5-12x}$

Step1: Simplify both sides

Simplify the left - hand side: $2\times10-2\times3x = 20-6x$.
Simplify the right - hand side: $6x-12x + 5=-6x + 5$.
So the equation becomes $20-6x=-6x + 5$.

Step2: Analyze the equation

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

Third Equation: $\boldsymbol{\frac{1}{2}(18x + 100)=20 - x}$

Step1: Distribute on the left - hand side

Using the distributive property $a(b + c)=ab+ac$, we have $\frac{1}{2}\times18x+\frac{1}{2}\times100 = 9x + 50$.
So the equation becomes $9x + 50=20 - x$.

Step2: Solve for x

Add $x$ to both sides: $9x+x + 50=20 - x+x$.
We get $10x+50 = 20$.
Subtract 50 from both sides: $10x+50 - 50=20 - 50$.
We get $10x=-30$.
Divide both sides by 10: $x=\frac{-30}{10}=-3$.

Answer:

s:

• For $\frac{2}{4}=\frac{6}{-2x + 8}$, $x=-2$
• For $2(10 - 3x)=6x + 5-12x$, No solution
• For $\frac{1}{2}(18x + 100)=20 - x$, $x=-3$