The distance between the points (1,3) and (x,7) is 5, find x.

Answer

To find the value of \( x \) such that the distance between the points \( (1, 3) \) and \( (x, 7) \) is 5, we can use the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Identify the points
Here, we have:
- Point 1: \( (x_1, y_1) = (1, 3) \)
- Point 2: \( (x_2, y_2) = (x, 7) \)

### Step 2: Set up the distance equation
According to the problem, the distance between these two points is 5. Therefore, we can write:

\[ \sqrt{(x - 1)^2 + (7 - 3)^2} = 5 \]

### Step 3: Simplify the equation
Now, simplify \( (7 - 3)^2 \):

\[ (7 - 3)^2 = 4^2 = 16 \]

So the equation becomes:

\[ \sqrt{(x - 1)^2 + 16} = 5 \]

### Step 4: Square both sides
To eliminate the square root, we square both sides of the equation:

\[ (x - 1)^2 + 16 = 5^2 \]

This simplifies to:

\[ (x - 1)^2 + 16 = 25 \]

### Step 5: Isolate the squared term
Next, we isolate \( (x - 1)^2 \) by subtracting 16 from both sides:

\[ (x - 1)^2 = 25 - 16 \]

This gives us:

\[ (x - 1)^2 = 9 \]

### Step 6: Take the square root
Now, we take the square root of both sides:

\[ x - 1 = \pm 3 \]

### Step 7: Solve for \( x \)
This gives us two equations to solve:

1. \( x - 1 = 3 \)
2. \( x - 1 = -3 \)

For the first equation:

\[ x = 3 + 1 = 4 \]

For the second equation:

\[ x = -3 + 1 = -2 \]

### Final Answer
Thus, the values of \( x \) are:

\[ x = 4 \quad \text{and} \quad x = -2 \]