Real-Life Situation
Imagine a teacher distributing packets of worksheets to students. Each packet contains a certain number of worksheets, and each student receives one packet. The number of worksheets in each packet is represented by \( x \). The teacher has a total of 144 worksheets, and we know that each packet contains \( 9 \) worksheets in addition to the base number of \( x \). The equation representing the total number of worksheets is:
\[ 8(x + 9) = 144 \]
This means there are 8 packets, and each contains \( x + 9 \) worksheets.
Solving the Equation
Step 1: Expand the Left Side
We start with:
\[ 8(x + 9) = 144 \]
Distributing \( 8 \) gives:
\[ 8x + 72 = 144 \]
Step 2: Isolate \( 8x \)
Subtract \( 72 \) from both sides:
\[ 8x = 144 - 72 \]
Calculating the right side gives:
\[ 8x = 72 \]
Step 3: Solve for \( x \)
Next, divide both sides by \( 8 \):
\[ x = \frac{72}{8} \]
Calculating this, we find:
\[ x = 9 \]
Alternate Solution Method: Substitution
Another way to solve the original equation is by substituting \( x \) directly into the expanded equation.
Starting again with:
\[ 8(x + 9) = 144 \]
We can directly isolate \( (x + 9) \) by dividing both sides by \( 8 \):
\[ x + 9 = \frac{144}{8} \]
Calculating the right side gives us:
\[ x + 9 = 18 \]
Now, we subtract \( 9 \) from both sides:
\[ x = 18 - 9 \]
Calculating this yields:
\[ x = 9 \]
Summary of Solutions
In both methods, we arrived at the same solution, \( x = 9 \).
Preferred Solution Method
I preferred the direct substitution method because it was more straightforward and required fewer steps. Reducing the equation right away simplified the process, making it easier to see the relationship between the variables without having to handle intermediate expressions. The substitution approach feels more intuitive and quicker to compute, which is beneficial in real-life situations where speed and accuracy are essential.