\begin{tabular}{|c|c|c|c|}
\hline Step & Statement & \multicolumn{2}{|c|}{Reason} \\
\hline 1 & [tex]$\frac{1}{4} x - \frac{9}{4} x - 7 = -15$[/tex] & Given & \\
\hline 2 & [tex]$-2x - 7 = -15$[/tex] & & Simplification \\
\hline 3 & [tex]$-2x - 7 + 7 = -15 + 7$[/tex] & & Addition Property of Equality \\
\hline 4 & [tex]$-2x = -8$[/tex] & & Simplification \\
\hline 5 & [tex]$\frac{-2x}{-2} = \frac{-8}{-2}$[/tex] & & Division Property of Equality \\
\hline 6 & [tex]$x = 4$[/tex] & & Simplification \\
\hline
\end{tabular}



Answer :

Certainly! Let's solve the equation [tex]\( \frac{1}{4} x - \frac{9}{4} x - 7 = -15 \)[/tex] step-by-step, thoroughly explaining each part of the process:

### Step-by-Step Solution:

1. Given Equation:
[tex]\[ \frac{1}{4} x - \frac{9}{4} x - 7 = -15 \][/tex]
We start with the given linear equation.

2. Combine Like Terms:
We need to combine the terms involving [tex]\( x \)[/tex]. Both [tex]\(\frac{1}{4} x\)[/tex] and [tex]\(\frac{9}{4} x\)[/tex] are like terms because they both contain [tex]\( x \)[/tex]:

[tex]\[ \frac{1}{4} x - \frac{9}{4} x \][/tex]
Calculate the coefficient of [tex]\( x \)[/tex]:

[tex]\[ \frac{1}{4} - \frac{9}{4} = -\frac{8}{4} = -2 \][/tex]
Thus, the equation reduces to:

[tex]\[ -2x - 7 = -15 \][/tex]

3. Add 7 to Both Sides:
Next, we isolate the term involving [tex]\( x \)[/tex] by eliminating the constant term [tex]\(-7\)[/tex] on the left-hand side. We do this by adding 7 to both sides:

[tex]\[ -2x - 7 + 7 = -15 + 7 \][/tex]

Simplifying both sides:

[tex]\[ -2x = -8 \][/tex]

4. Divide by the Coefficient of x:
The last step is to solve for [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\(-2\)[/tex]:

[tex]\[ \frac{-2x}{-2} = \frac{-8}{-2} \][/tex]

Simplifying the division:

[tex]\[ x = 4 \][/tex]

Therefore, the solution to the equation [tex]\( \frac{1}{4} x - \frac{9}{4} x - 7 = -15 \)[/tex] is:

[tex]\[ \boxed{4} \][/tex]