Find the distance between the following pairs of points :
(-5,7), (-1,3)

Answer

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

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

### Step 1: Identify the coordinates
Let:
- Point 1: \((x_1, y_1) = (-5, 7)\)
- Point 2: \((x_2, y_2) = (-1, 3)\)

### Step 2: Substitute the coordinates into the distance formula
Substituting the values into the formula:

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

### Step 3: Simplify the expressions inside the parentheses
Calculating the differences:

\[ x_2 - x_1 = -1 + 5 = 4 \]
\[ y_2 - y_1 = 3 - 7 = -4 \]

### Step 4: Substitute the simplified differences back into the formula
Now substituting these values back into the distance formula:

\[ d = \sqrt{(4)^2 + (-4)^2} \]

### Step 5: Calculate the squares
Calculating the squares:

\[ d = \sqrt{16 + 16} \]

### Step 6: Add the squares
Adding the squares together:

\[ d = \sqrt{32} \]

### Step 7: Simplify the square root
We can simplify \(\sqrt{32}\):

\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

### Final Answer
Thus, the distance between the points (-5, 7) and (-1, 3) is:

\[ d = 4\sqrt{2} \text{ units} \]

---