Question

What is the equation for a line perpendicular to \(6 x-7 y=8\) and passing through the point \((8,1)\) ? A. \(y=-\frac{7}{6} x+\frac{31}{3}\) B. \(y=\frac{6}{7} x-\frac{8}{7}\) C. \(-6 x+7 y=-8\) D. \(y=\frac{7}{6} x-\frac{31}{3}\)

Short answer

The correct answer is A: \( y = -\frac{7}{6}x + \frac{31}{3}\)

Step-by-step solution

Identify the slope of the given line

The given line is in the form of the linear equation: \[6x - 7y = 8\]To find the slope, rewrite it in slope-intercept form (\(y = mx + b\)). Start by isolating \(y\): \[-7y = -6x + 8\] Divide by \(-7\):\[y = \frac{6}{7}x - \frac{8}{7}\]The slope of the given line, \(m_1\), is \(\frac{6}{7}\).

Find the slope of the perpendicular line

For a line to be perpendicular to another, its slope \(m_2\) must be the negative reciprocal of the slope of the original line.Since the slope of the original line is \(\frac{6}{7}\), the slope of the perpendicular line is:\[m_2 = -\frac{7}{6}\]

Use the slope and point to find the equation

The perpendicular line passes through the point \((8, 1)\) and has a slope of \(-\frac{7}{6}\). Use the point-slope form of the equation: \[y - y_1 = m(x - x_1)\]Substitute \(m = -\frac{7}{6}\), \(x_1 = 8\), and \(y_1 = 1\): \[y - 1 = -\frac{7}{6}(x - 8)\]

Simplify the equation

Distribute the slope on the right side: \[y - 1 = -\frac{7}{6}x + \frac{56}{6}\]Simplify the constant term: \[y - 1 = -\frac{7}{6}x + \frac{28}{3}\] Finally, add 1 to both sides:\[y = -\frac{7}{6}x + \frac{28}{3} + 1\]Convert 1 to a fraction over 3:\[y = -\frac{7}{6}x + \frac{28}{3} + \frac{3}{3}\]Combine like terms:\[y = -\frac{7}{6}x + \frac{31}{3}\]

Verify the correct answer

The equation of the line perpendicular to \(6x - 7y = 8\) and passing through the point \((8, 1)\) is: \[ y = -\frac{7}{6}x + \frac{31}{3}\]Therefore, the correct answer is A: \( y = -\frac{7}{6}x + \frac{31}{3}\)

Key concepts

slope-intercept form

The slope-intercept form is one of the most common ways to write the equation of a line. It's expressed as
\( y = mx + b \), where:

• \( y \) is the dependent variable (output).
• \( x \) is the independent variable (input).
• \( m \) is the slope of the line.
• \( b \) is the y-intercept (the point where the line crosses the y-axis).

To rewrite a linear equation in this form, we isolate \( y \). For example, given the equation \( 6x - 7y = 8 \), we solve for \( y \) to get:
\( -7y = -6x + 8 \)
Divide everything by \( -7 \) to get the slope-intercept form:
\( y = \frac{6}{7}x - \frac{8}{7} \)
Here, the slope \( m \) is \( \frac{6}{7} \) and the y-intercept \( b \) is \( -\frac{8}{7} \).

negative reciprocal

When it comes to perpendicular lines, their slopes have a special relationship. Specifically, the slope of one line is the negative reciprocal of the slope of the other. This relationship ensures 90-degree angles between the two lines.
If the slope of the original line is \( m \), then the slope of the perpendicular line is \( -\frac{1}{m} \). In our example, the slope of the given line is \( \frac{6}{7} \). Therefore, the slope of the perpendicular line is:
\( -\frac{1}{ \frac{6}{7} } \)
When you take the reciprocal of \( \frac{6}{7} \), you get \( \frac{7}{6} \), and negating it gives \( -\frac{7}{6} \).
So, the slope of the perpendicular line to the given line is \( -\frac{7}{6} \). Knowing this is crucial for finding the perpendicular line’s equation. This step is essential and ensures the perpendicularity of the new line.

point-slope form

The point-slope form is another valuable method for writing the equation of a line, especially when you know a point on the line and its slope. It's expressed as:
\( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a point on the line.
• \( m \) is the slope of the line.

Using the example from the exercise, we have the point \( (8, 1) \) and the slope \( -\frac{7}{6} \) from the negative reciprocal. Plugging these into the point-slope formula, we get:
\( y - 1 = -\frac{7}{6}(x - 8) \)
This form lets you quickly write the equation of a line when a single point and the slope are known. To convert it into slope-intercept form, distribute the slope and simplify:
\( y - 1 = -\frac{7}{6}x + \frac{56}{6} \)
\( y - 1 = -\frac{7}{6}x + \frac{28}{3} \)
Add 1 to both sides to isolate \( y \):
\( y = -\frac{7}{6}x + \frac{28}{3} + 1 \)
Convert \( 1 \) to a fraction over 3 to combine like terms:
\( y = -\frac{7}{6}x + \frac{31}{3} \)
This final equation matches option A, verifying our solution.