Find the equation of the line which passes through (2, 7) and whose y-intercept is 3.

Answer

To find the equation of the line that passes through the point (2, 7) and has a y-intercept of 3, we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point P (2, 7) which is given.
- Point Q (0, 3) which represents the y-intercept (where x = 0 and y = 3).

### Step 2: Calculate the Slope
The slope (m) of the line can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can assign:
- \( (x_1, y_1) = (2, 7) \) (Point P)
- \( (x_2, y_2) = (0, 3) \) (Point Q)

Substituting the values into the slope formula:
\[ m = \frac{3 - 7}{0 - 2} = \frac{-4}{-2} = 2 \]

### Step 3: Use the Point-Slope Form of the Equation
Now that we have the slope (m = 2) and a point (2, 7), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = 2\), \(x_1 = 2\), and \(y_1 = 7\):
\[ y - 7 = 2(x - 2) \]

### Step 4: Simplify the Equation
Now we simplify the equation:
\[ y - 7 = 2x - 4 \]
Adding 7 to both sides:
\[ y = 2x - 4 + 7 \]
\[ y = 2x + 3 \]

### Step 5: Rearranging to Standard Form
To express this in standard form \(Ax + By + C = 0\):
\[ y - 2x - 3 = 0 \]
or rearranging gives:
\[ -2x + y - 3 = 0 \]
which can also be written as:
\[ 2x - y + 3 = 0 \]

### Final Equation
Thus, the equation of the line is:
\[ 2x - y + 3 = 0 \]

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