Find the equation of a line :
passing through the points (-3, 1) and (1,5).

Answer

To find the equation of the line passing through the points (-3, 1) and (1, 5), we will follow these steps:

### Step 1: Identify the coordinates of the points
Let the points be:
- Point A (x1, y1) = (-3, 1)
- Point B (x2, y2) = (1, 5)

### Step 2: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of points A and B into the formula:
- \( y_1 = 1 \)
- \( y_2 = 5 \)
- \( x_1 = -3 \)
- \( x_2 = 1 \)

We get:
\[ y - 1 = \frac{5 - 1}{1 - (-3)} (x - (-3)) \]

### Step 4: Simplify the equation
Calculate the slope:
- \( y_2 - y_1 = 5 - 1 = 4 \)
- \( x_2 - x_1 = 1 - (-3) = 1 + 3 = 4 \)

So, the equation becomes:
\[ y - 1 = \frac{4}{4} (x + 3) \]
\[ y - 1 = 1(x + 3) \]
\[ y - 1 = x + 3 \]

### Step 5: Rearrange to standard form
Now, rearranging the equation:
\[ y - x = 3 + 1 \]
\[ y - x = 4 \]

### Final Equation
Thus, the equation of the line passing through the points (-3, 1) and (1, 5) is:
\[ y - x = 4 \]
or equivalently,
\[ y = x + 4 \]

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