Find the equation of a line passes through the points `(3,4)` and parallel to the line `x+5y=1`.

Answer

To find the equation of a line that passes through the point (3, 4) and is parallel to the line given by the equation \(x + 5y = 1\), we can follow these steps:

### Step 1: Find the slope of the given line
The first step is to rewrite the equation \(x + 5y = 1\) in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. Start with the original equation:
\[ x + 5y = 1 \]
2. Rearrange to solve for \(y\):
\[ 5y = -x + 1 \]
3. Divide by 5:
\[ y = -\frac{1}{5}x + \frac{1}{5} \]

From this, we can see that the slope \(m\) of the line is \(-\frac{1}{5}\).

### Step 2: Use the point-slope form to find the equation of the new line
Since the new line is parallel to the given line, it will have the same slope. We can use the point-slope form of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.

1. Substitute \(m = -\frac{1}{5}\), \(x_1 = 3\), and \(y_1 = 4\):
\[ y - 4 = -\frac{1}{5}(x - 3) \]

### Step 3: Simplify the equation
Now, we will simplify the equation obtained from the point-slope form.

1. Distribute the slope on the right side:
\[ y - 4 = -\frac{1}{5}x + \frac{3}{5} \]
2. Add 4 to both sides:
\[ y = -\frac{1}{5}x + \frac{3}{5} + 4 \]
3. Convert 4 into a fraction with a denominator of 5:
\[ 4 = \frac{20}{5} \]
4. Combine the fractions:
\[ y = -\frac{1}{5}x + \frac{3}{5} + \frac{20}{5} = -\frac{1}{5}x + \frac{23}{5} \]

### Step 4: Convert to standard form
To express the equation in standard form \(Ax + By + C = 0\):

1. Multiply through by 5 to eliminate the fraction:
\[ 5y = -x + 23 \]
2. Rearranging gives:
\[ x + 5y - 23 = 0 \]

Thus, the equation of the line that passes through the point (3, 4) and is parallel to the line \(x + 5y = 1\) is:
\[ x + 5y - 23 = 0 \]