Solve the following equations and verify the results.
`2/3(t-5)-1/4(t-2)=1`

Answer

To solve the equation \( \frac{2}{3}(t-5) - \frac{1}{4}(t-2) = 1 \), we will follow these steps:

### Step 1: Distribute the fractions
Distributing the fractions on the left side of the equation gives us:
\[ \frac{2}{3}t - \frac{10}{3} - \frac{1}{4}t + \frac{1}{2} = 1 \]

### Step 2: Find a common denominator
The least common multiple (LCM) of the denominators (3 and 4) is 12. We will multiply each term by 12 to eliminate the fractions:
\[ 12 \left(\frac{2}{3}t\right) - 12 \left(\frac{10}{3}\right) - 12 \left(\frac{1}{4}t\right) + 12 \left(\frac{1}{2}\right) = 12 \cdot 1 \]
This simplifies to:
\[ 8t - 40 - 3t + 6 = 12 \]

### Step 3: Combine like terms
Now, combine the \(t\) terms and the constant terms:
\[ (8t - 3t) + (-40 + 6) = 12 \]
This simplifies to:
\[ 5t - 34 = 12 \]

### Step 4: Isolate the variable
Next, we will isolate \(t\) by adding 34 to both sides:
\[ 5t = 12 + 34 \]
This simplifies to:
\[ 5t = 46 \]

### Step 5: Solve for \(t\)
Now, divide both sides by 5:
\[ t = \frac{46}{5} \]

### Step 6: Verification
To verify, substitute \(t = \frac{46}{5}\) back into the original equation:
\[ \frac{2}{3}\left(\frac{46}{5} - 5\right) - \frac{1}{4}\left(\frac{46}{5} - 2\right) = 1 \]
Calculating each term:
1. \( \frac{46}{5} - 5 = \frac{46}{5} - \frac{25}{5} = \frac{21}{5} \)
2. \( \frac{2}{3} \cdot \frac{21}{5} = \frac{42}{15} = \frac{14}{5} \)
3. \( \frac{46}{5} - 2 = \frac{46}{5} - \frac{10}{5} = \frac{36}{5} \)
4. \( \frac{1}{4} \cdot \frac{36}{5} = \frac{36}{20} = \frac{9}{5} \)

Now substituting back into the equation:
\[ \frac{14}{5} - \frac{9}{5} = \frac{5}{5} = 1 \]
Since both sides are equal, our solution is verified.

### Final Answer
Thus, the solution to the equation is:
\[ t = \frac{46}{5} \]