Question

T
W
L
If WT = 53 and LW = 45, find TL. You must show all work

Answer 1

To find \( TL \) given \( WT = 53 \) and \( LW = 45 \), we can use the relationships we have.

Let's denote the values of each variable as follows:
- \( T \) for \( T \)
- \( W \) for \( W \)
- \( L \) for \( L \)

From the equations given, we know:
1. \( W \cdot T = 53 \) (equation 1)
2. \( L \cdot W = 45 \) (equation 2)

We want to find \( T \cdot L \) (equation 3).

### Step 1: Solve for \( W \)

From equation 1, we can express \( W \) in terms of \( T \):
\[ W = \frac{53}{T} \]

### Step 2: Substitute \( W \) into equation 2

Now we substitute \( W \) from equation 1 into equation 2:

\[
L \cdot \left( \frac{53}{T} \right) = 45
\]

### Step 3: Rearranging the equation

Multiply both sides by \( T \) to eliminate the fraction:
\[
L \cdot 53 = 45T
\]

Now we can solve for \( L \):
\[
L = \frac{45T}{53}
\]

### Step 4: Finding \( T \cdot L \)

Now we find \( T \cdot L \) using the expression we derived:
\[
T \cdot L = T \cdot \left( \frac{45T}{53} \right)
\]

This simplifies to:
\[
T \cdot L = \frac{45T^2}{53}
\]

### Step 5: Express \( TL \) in simpler terms

Since we don't have a specific numerical value for \( T \), \( TL \) in terms of \( T \) is expressed as:
\[
TL = \frac{45T^2}{53}
\]

Unfortunately, without an additional relationship or more information about \( T \) or \( L \), we cannot find a specific numerical value for \( TL \) directly.

However, we can check if there's a way to eliminate \( T \) or to calculate \( TL \) using the equations derived.

To compute \( TL \):
- If we multiply the product of the two equations together:
\[
(WT) \cdot (LW) = (TL)(W)
\]

Focusing on \( TL \):
\[
(WT)(LW) = 53 \cdot 45
\]

\[
TL \cdot W = 53 \cdot 45
\]

Now to find \( W \):
- Using the expression for \( W \):
\[
TL \cdot \left(\frac{53}{T}\right) = 53 \cdot 45
\]

### Conclusion

Thus, we arrive at:
\[
TL = \frac{53 \cdot 45 \cdot T}{53}
\]
This gives:
\[
TL = 45T
\]
At this point, without loss of generality, we derived the result \( TL \) expressed in terms of \( T \). However, the exercise does not give a specific numerical value, confirming once again that direct computation of \( TL \) theoretically leads us to a relationship dependent on \( T \) without identifiable actual values.

Since the initial problem doesn't provide extra conditions or values, a complete numeric answer for \( TL \) is not quantifiable under current details in this approach, but the derived relationship suffices for analytical perspectives.