The distance betweent the points (3,1) and (0,x) is 5. Find x.

Answer

To solve the problem of finding the value of \( x \) such that the distance between the points \( (3, 1) \) and \( (0, x) \) is 5, we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
In our case, the points are \( (3, 1) \) and \( (0, x) \).

### Step 2: Substitute the Points into the Formula
Here, \( x_1 = 3 \), \( y_1 = 1 \), \( x_2 = 0 \), and \( y_2 = x \). Substituting these values into the distance formula gives:
\[ d = \sqrt{(3 - 0)^2 + (1 - x)^2} \]
This simplifies to:
\[ d = \sqrt{3^2 + (1 - x)^2} = \sqrt{9 + (1 - x)^2} \]

### Step 3: Set the Distance Equal to 5
According to the problem, the distance is given as 5. Therefore, we can set up the equation:
\[ \sqrt{9 + (1 - x)^2} = 5 \]

### Step 4: Square Both Sides to Eliminate the Square Root
To eliminate the square root, we square both sides of the equation:
\[ 9 + (1 - x)^2 = 25 \]

### Step 5: Solve for \( (1 - x)^2 \)
Now, we can isolate \( (1 - x)^2 \):
\[ (1 - x)^2 = 25 - 9 \]
\[ (1 - x)^2 = 16 \]

### Step 6: Take the Square Root of Both Sides
Taking the square root of both sides gives us:
\[ 1 - x = \pm 4 \]

### Step 7: Solve for \( x \)
Now we will solve for \( x \) in both cases:
1. **Case 1**: \( 1 - x = 4 \)
\[ -x = 4 - 1 \]
\[ -x = 3 \implies x = -3 \]

2. **Case 2**: \( 1 - x = -4 \)
\[ -x = -4 - 1 \]
\[ -x = -5 \implies x = 5 \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = -3 \quad \text{and} \quad x = 5 \]

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