Certainly! Let's solve this step-by-step.
The given equation is:
[tex]\[ 10 \left( \frac{3a}{5} - y \right) = \frac{3a}{2} - \frac{9}{2} \][/tex]
### Step 1: Distribute the 10 inside the parenthesis
First, distribute the 10 to both terms inside the parenthesis on the left-hand side:
[tex]\[ 10 \cdot \frac{3a}{5} - 10y = \frac{3a}{2} - \frac{9}{2} \][/tex]
### Step 2: Simplify the multiplication
Simplify the term [tex]\(10 \cdot \frac{3a}{5}\)[/tex]:
[tex]\[ 10 \cdot \frac{3a}{5} = 6a \][/tex]
So, the equation now becomes:
[tex]\[ 6a - 10y = \frac{3a}{2} - \frac{9}{2} \][/tex]
### Step 3: Clear the fractions
To eliminate the fractions on the right-hand side, multiply every term in the equation by 2:
[tex]\[ 2(6a) - 2(10y) = 2 \left( \frac{3a}{2} \right) - 2 \left( \frac{9}{2} \right) \][/tex]
This simplifies to:
[tex]\[ 12a - 20y = 3a - 9 \][/tex]
### Step 4: Rearrange terms to isolate [tex]\(y\)[/tex]
Subtract [tex]\(3a\)[/tex] from both sides to get all [tex]\(a\)[/tex]-terms on one side:
[tex]\[ 12a - 3a - 20y = -9 \][/tex]
Simplify the left-hand side:
[tex]\[ 9a - 20y = -9 \][/tex]
### Step 5: Solve for [tex]\(y\)[/tex]
Add [tex]\(20y\)[/tex] to both sides:
[tex]\[ 9a = 20y - 9 \][/tex]
Next, add 9 to both sides:
[tex]\[ 9a + 9 = 20y \][/tex]
Divide everything by 20 to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{9a + 9}{20} \][/tex]
You can factor out a 9 from the numerator:
[tex]\[ y = \frac{9(a + 1)}{20} \][/tex]
Simplify the fraction:
[tex]\[ y = \frac{9}{20}a + \frac{9}{20} \][/tex]
Finally, since [tex]\(\frac{9}{20}\)[/tex] simplifies to 0.45, we get:
[tex]\[ y = 0.45a + 0.45 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ y = 0.45a + 0.45 \][/tex]
