Solve:
$\frac{2 a-5}{4}-\frac{1-a}{3}=\frac{3 a+1}{6}$
To solve the linear equation \[\frac{2a - 5}{4} - \frac{1 - a}{3} = \frac{3a + 1}{6},\] we first identify the denominators 4, 3, and 6. To simplify the equation, we determine the least common multiple (LCM) of these denominators, which is 12. Multiplying every term in the equation by 12 will eliminate the fractions.
Multiplying both sides of the equation by 12 yields:
\[12 \cdot \left(\frac{2a - 5}{4}\right) - 12 \cdot \left(\frac{1 - a}{3}\right) = 12 \cdot \left(\frac{3a + 1}{6}\right).\]
Simplifying each term gives:
\[3(2a - 5) - 4(1 - a) = 2(3a + 1).\]
Next, we expand the expressions:
\[6a - 15 - 4 + 4a = 6a + 2.\]
Combining like terms on the left side, we get:
\[10a - 19 = 6a + 2.\]
Now, we isolate the variable \(a\). Subtracting \(6a\) from both sides:
\[4a - 19 = 2.\]
Adding 19 to both sides:
\[4a = 21.\]
Finally, dividing by 4:
\[a = \frac{21}{4} = 5.25.\]
Thus, the solution is \(a = 5.25\).
Linear equations like this are foundational for solving real-world problems involving rates and proportions. For example, imagine you are managing a project budget where costs are split across different departments based on fractional shares. If you need to determine the specific allocation amount represented by a variable that satisfies a budget balance equation, you use algebraic techniques just like this to find the exact value needed to maintain financial equality. Understanding these steps ensures you can solve for unknown variables in any proportional resource allocation scenario.