Let's work through the problem step by step.
We are given certain expressions and need to perform algebraic manipulations to find the value of the variable [tex]\( a \)[/tex].
### Step-by-Step Solution
1. Define Variables:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 9 \)[/tex]
2. Form Expressions:
- Left Expression: [tex]\( 18 - 3a \)[/tex]
- Right Expression: [tex]\( b(a + 1) \)[/tex]
3. Calculate Each Side Separately:
- Substitute [tex]\( a \)[/tex] into the left expression:
[tex]\[
\text{Left Side} = a \cdot (18 - 3a)
\][/tex]
Substitute [tex]\( a = 4 \)[/tex]:
[tex]\[
\text{Left Side} = 4 \cdot (18 - 3 \cdot 4) = 4 \cdot 6 = 24
\][/tex]
- Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the right expression:
[tex]\[
\text{Right Side} = b \cdot (a + 1)
\][/tex]
Substitute [tex]\( b = 9 \)[/tex] and [tex]\( a = 4 \)[/tex]:
[tex]\[
\text{Right Side} = 9 \cdot (4 + 1) = 9 \cdot 5 = 45
\][/tex]
4. Combine Like Terms:
- Set up the equation:
[tex]\[
72 - 12a - (9a + 9) = 0
\][/tex]
Simplify:
[tex]\[
72 - 12a - 9a - 9 = 0
\][/tex]
Combine like terms:
[tex]\[
72 - 21a - 9 = 0
\][/tex]
Further simplify:
[tex]\[
63 - 21a = 0
\][/tex]
5. Solve for [tex]\( a \)[/tex]:
- Isolate [tex]\( a \)[/tex]:
[tex]\[
-21a + 63 = 0
\][/tex]
Move the constant term to the right side:
[tex]\[
-21a = -63
\][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{-63}{-21} = 3
\][/tex]
### Conclusion
We determined that the value for [tex]\( a \)[/tex] is [tex]\( 3 \)[/tex].
