Find an equation for the line passing through the point (2, 1) and perpendicular to the line 
y=5x-14

To find the equation of the line that passes through the point \((2, 1)\) and is perpendicular to the line \(y = 5x - 14\), follow these steps:

### Step 1: Find the slope of the given line
The equation of the line is \(y = 5x - 14\). This is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. So, the slope of the given line is:

\[
m_1 = 5
\]

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the slope of the perpendicular line, \(m_2\), is:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{5}
\]

### Step 3: Use the point-slope form to find the equation of the line
Now, we know the slope of the perpendicular line is \(-\frac{1}{5}\), and it passes through the point \((2, 1)\). Using the point-slope form of a line equation:

\[
y - y_1 = m(x - x_1)
\]

Substitute \(m = -\frac{1}{5}\), \(x_1 = 2\), and \(y_1 = 1\):

\[
y - 1 = -\frac{1}{5}(x - 2)
\]

### Step 4: Simplify the equation
Expand and simplify the equation:

\[
y - 1 = -\frac{1}{5}x + \frac{2}{5}
\]
\[
y = -\frac{1}{5}x + \frac{2}{5} + 1
\]
\[
y = -\frac{1}{5}x + \frac{2}{5} + \frac{5}{5}
\]
\[
y = -\frac{1}{5}x + \frac{7}{5}
\]

### Final Equation:
The equation of the line passing through \((2, 1)\) and perpendicular to the line \(y = 5x - 14\) is:

\[
y = -\frac{1}{5}x + \frac{7}{5}
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you find the equation of a line parallel to a given line?
2. What is the geometric interpretation of perpendicular slopes?
3. How do you convert a line equation from point-slope form to slope-intercept form?
4. What is the standard form of a linear equation?
5. How do you determine if two lines are perpendicular based on their equations?

**Tip:** To find the perpendicular slope, always take the negative reciprocal of the original line's slope.

Find an equation for the line passing through the point (2, 1) and perpendicular to the line y = 5x - 14.

Discover how to find the equation of a line that passes through the point (2, 1) and is perpendicular to y = 5x - 14. This solution covers key concepts such as slope-intercept form, point-slope form, and the relationship between perpendicular slopes. Ideal for students in Grades 8-10.

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