Sure, let's provide detailed justifications for each step in the solution to the given equation.
1. \(\frac{1}{4} x - \frac{9}{4} x - 7 = -15\)
- Statement: This is the original equation given.
2. \(-2 x - 7 = -15\)
- Justification: Combine the like terms \(\frac{1}{4} x\) and \(\frac{9}{4} x\). Since \(\frac{1}{4} x - \frac{9}{4} x = -2 x\), the equation simplifies to \(-2 x - 7 = -15\).
3. \(-2 x - 7 + 7 = -15 + 7\)
- Justification: Add 7 to both sides of the equation to isolate the term with \(x\). This is a step to get rid of the constant term on the left side.
4. \(-2 x = -8\)
- Justification: Simplify both sides of the equation after adding 7 to \(-15\) gives \(-8\).
5. \(\frac{-2 x}{-2} = \frac{-8}{-2}\)
- Justification: Divide both sides of the equation by \(-2\) to solve for \(x\).
6. \(x = 4\)
- Justification: Simplify \(\frac{-8}{-2}\) to get \(x = 4\).
So each step follows logically from the previous by applying basic algebraic principles of combining like terms, isolating the variable term, and solving for the variable.
