`|t-7|=4`
`|9-t|=2`
What value of r satisfies both of the above equations?

Answer

To solve the equations \( |t-7|=4 \) and \( |9-t|=2 \), we will break them down step by step.

### Step 1: Solve the first equation \( |t-7|=4 \)

The absolute value equation \( |t-7|=4 \) can be split into two cases:

**Case 1:**
\[ t - 7 = 4 \]
To solve for \( t \):
\[ t = 4 + 7 = 11 \]

**Case 2:**
\[ t - 7 = -4 \]
To solve for \( t \):
\[ t = -4 + 7 = 3 \]

So, the solutions for the first equation \( |t-7|=4 \) are \( t = 11 \) and \( t = 3 \).

### Step 2: Solve the second equation \( |9-t|=2 \)

Similarly, we will split the absolute value equation \( |9-t|=2 \) into two cases:

**Case 1:**
\[ 9 - t = 2 \]
To solve for \( t \):
\[ -t = 2 - 9 \]
\[ -t = -7 \implies t = 7 \]

**Case 2:**
\[ 9 - t = -2 \]
To solve for \( t \):
\[ -t = -2 - 9 \]
\[ -t = -11 \implies t = 11 \]

So, the solutions for the second equation \( |9-t|=2 \) are \( t = 7 \) and \( t = 11 \).

### Step 3: Find the common solution

Now we have the solutions from both equations:
- From \( |t-7|=4 \): \( t = 11, 3 \)
- From \( |9-t|=2 \): \( t = 7, 11 \)

The common solution between these two sets is:
\[ t = 11 \]

### Final Answer:
The value of \( t \) that satisfies both equations is \( t = 11 \).

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