Find the equation of a line passing through point (5, 1) and parallel to the line 7x – 2y + 5 = 0.

Answer

To find the equation of a line passing through the point (5, 1) and parallel to the line given by the equation \(7x - 2y + 5 = 0\), we can follow these steps:

### Step 1: Determine the slope of the given line
To find the slope of the line \(7x - 2y + 5 = 0\), we need to rewrite it in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

Starting with the equation:
\[ 7x - 2y + 5 = 0 \]
we can rearrange it to solve for \(y\):
\[ -2y = -7x - 5 \]
Dividing everything by -2 gives:
\[ y = \frac{7}{2}x + \frac{5}{2} \]
From this, we can see that the slope \(m\) of the given line is:
\[ m = \frac{7}{2} \]

### Step 2: Use the slope for the parallel line
Since parallel lines have the same slope, the slope of the line we want to find will also be:
\[ m = \frac{7}{2} \]

### Step 3: Use the point-slope form of the line equation
We will use the point-slope form of the line equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes. Here, \((x_1, y_1) = (5, 1)\).

Substituting the values into the equation:
\[ y - 1 = \frac{7}{2}(x - 5) \]

### Step 4: Simplify the equation
Now, we will simplify this equation:
\[ y - 1 = \frac{7}{2}x - \frac{35}{2} \]
Adding 1 (or \(\frac{2}{2}\)) to both sides gives:
\[ y = \frac{7}{2}x - \frac{35}{2} + \frac{2}{2} \]
\[ y = \frac{7}{2}x - \frac{33}{2} \]

### Step 5: Convert to standard form
To convert this to standard form \(Ax + By + C = 0\), we can rearrange it:
\[ \frac{7}{2}x - y - \frac{33}{2} = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 7x - 2y - 33 = 0 \]

Thus, the equation of the line passing through the point (5, 1) and parallel to the line \(7x - 2y + 5 = 0\) is:
\[ 7x - 2y - 33 = 0 \]