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]
