We are tasked with solving the equation:

\[
\frac{9}{8}(4 - \frac{1}{2}a) = 3\left(\frac{1}{4}a + \frac{5}{2}\right)
\]

### Step 1: Distribute the constants on both sides

On the left-hand side, distribute \(\frac{9}{8}\):

\[
\frac{9}{8} \cdot 4 - \frac{9}{8} \cdot \frac{1}{2}a = \frac{36}{8} - \frac{9}{16}a = \frac{9}{2} - \frac{9}{16}a
\]

On the right-hand side, distribute 3:

\[
3 \cdot \frac{1}{4}a + 3 \cdot \frac{5}{2} = \frac{3}{4}a + \frac{15}{2}
\]

Now, the equation is:

\[
\frac{9}{2} - \frac{9}{16}a = \frac{3}{4}a + \frac{15}{2}
\]

### Step 2: Eliminate fractions by multiplying the entire equation by 16 (the least common denominator):

\[
16 \left(\frac{9}{2} - \frac{9}{16}a \right) = 16 \left(\frac{3}{4}a + \frac{15}{2} \right)
\]

Simplifying:

\[
72 - 9a = 12a + 120
\]

### Step 3: Combine like terms

Move all terms with \(a\) on one side and constants on the other:

\[
72 - 120 = 12a + 9a
\]

\[
-48 = 21a
\]

### Step 4: Solve for \(a\)

\[
a = \frac{-48}{21} = -\frac{16}{7}
\]

Thus, the solution is:

\[
a = -\frac{16}{7}
\]

Would you like further details or have any questions? Here are some related questions for practice:

1. Solve \( \frac{5}{3}(6 - 2x) = 4(x + 3) \).
2. Simplify \( \frac{7}{2}(3 - \frac{1}{4}y) = 5(y + \frac{5}{3}) \).
3. Solve for \(b\): \( \frac{8}{5}(b - \frac{1}{2}) = 3(b + 4) \).
4. Expand and simplify \( \frac{3}{4}(2 - a) = 5(\frac{2}{3}a + 1) \).
5. Solve the linear equation \( 6(x - \frac{1}{3}) = 4(x + 2) \).

**Tip:** Always check for common denominators when fractions are involved—it simplifies the process!

Discover how to solve the linear equation 9/8(4 - 1/2a) = 3(1/4a + 5/2) with a step-by-step guide. This example covers key algebraic concepts like fraction operations, distributive property, and combining like terms, suitable for students in Grades 8-10.

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