Find the equation of a line passing through point (5, 1) and parallel to the line 7x – 2y + 5 = 0.
(a)7x-2y-33=0
(b)6x-8y=0
(c)5x-8y-9=0
(d)None of these

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 will follow these steps:

### Step 1: Find the slope of the given line
First, we need to rewrite the equation of the given line in slope-intercept form (y = mx + b), where m is the slope.

Starting with the equation:
\[ 7x - 2y + 5 = 0 \]
Rearranging it to solve for y:
\[ -2y = -7x - 5 \]
Dividing by -2:
\[ y = \frac{7}{2}x + \frac{5}{2} \]

From this, we can see that the slope (m) of the given line is:
\[ m = \frac{7}{2} \]

### Step 2: Use the slope-point form to find the equation of the new line
Since we want a line that is parallel to the given line, it will have the same slope. We will use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes. Here, \((x_1, y_1) = (5, 1)\) and \(m = \frac{7}{2}\).

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

### Step 3: Simplify the equation
Now, we will simplify the equation:
\[ y - 1 = \frac{7}{2}x - \frac{35}{2} \]
Adding 1 (which is \(\frac{2}{2}\)) to both sides:
\[ y = \frac{7}{2}x - \frac{35}{2} + \frac{2}{2} \]
\[ y = \frac{7}{2}x - \frac{33}{2} \]

### Step 4: Convert to standard form
To convert this equation to standard form \(Ax + By + C = 0\), we will eliminate the fraction by multiplying through by 2:
\[ 2y = 7x - 33 \]
Rearranging gives:
\[ 7x - 2y - 33 = 0 \]

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