Solve for a:
$\begin{aligned} -\frac{1}{4}+\frac{3}{4} a & =-\left(-\frac{7}{4} a-1\right) \end{aligned}$
The objective is to solve the linear equation:
\[-\frac{1}{4} + \frac{3}{4}a = -\left(-\frac{7}{4}a - 1\right)\]
We need to isolate the variable \(a\) by expanding the right side, grouping the terms containing \(a\), and simplifying the constants.
First, distribute the negative sign on the right side of the equation:
\[-\left(-\frac{7}{4}a - 1\right) = \frac{7}{4}a + 1\]
Substituting this back into the original equation gives:
\[-\frac{1}{4} + \frac{3}{4}a = \frac{7}{4}a + 1\]
Next, move the terms containing \(a\) to the right side and the constants to the left side:
\[-\frac{1}{4} - 1 = \frac{7}{4}a - \frac{3}{4}a\]
Simplify the constant terms on the left:
\[-\frac{1}{4} - \frac{4}{4} = -\frac{5}{4}\]
Simplify the variable terms on the right:
\[\frac{7}{4}a - \frac{3}{4}a = \frac{4}{4}a = a\]
Thus, we have:
\[a = -\frac{5}{4}\]
To verify, substitute \(a = -\frac{5}{4}\) back into the original equation:
LHS: \(-\frac{1}{4} + \frac{3}{4}\left(-\frac{5}{4}\right) = -\frac{1}{4} - \frac{15}{16} = -\frac{4}{16} - \frac{15}{16} = -\frac{19}{16}\)
RHS: \(-\left(-\frac{7}{4}\left(-\frac{5}{4}\right) - 1\right) = -\left(\frac{35}{16} - 1\right) = -\left(\frac{35}{16} - \frac{16}{16}\right) = -\left(\frac{19}{16}\right) = -\frac{19}{16}\)
Since both sides match, the solution is correct. The final value is \(\boxed{-\frac{5}{4}}\).
Linear equations are fundamental in balancing simple budgets or resource allocations. For example, if you are tracking the balance of a personal expense account where initial costs are partially offset by returns, you might use an equation of this form to determine the specific rate or quantity needed to reach a zero-balance or a specific target. Solving for variables in this way helps in predicting future balances based on current patterns.