Question

Write an equation of the line through the point (a) parallel to the given line and (b) perpendicular to the given line. $$ \frac{\text { Point }}{(-6,4)} \quad \frac{\text { Line }}{3 x+4 y=7} $$

Short answer

The equation of the line through the point (-6,4) and parallel to the line 3x+4y=7 is \(y = -3/4x -2\). The equation of the line through the point (-6,4) and perpendicular to the line 3x+4y=7 is \(y = 4/3x + 6\).

Step-by-step solution

Slope of the Given Line

Start by finding the slope of the given line. Convert the equation of the line \(3x+4y=7\) into slope-intercept form \(y=mx+b\) where \(m\) is the slope. The slope of a line in the form \(ax + by = c\) is \(-a/b\). Thus, the slope of the line is \(-3/4\).

Equation Parallel to the Given Line

The equation of the line parallel to the given line through the point \(-6,4\) will have the same slope as the given line, which is \(-3/4\). Using the slope-point form \(y-y_1=m(x-x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line, substitute \(-3/4\) as \(m\), \(-6\) as \(x_1\) and \(4\) as \(y_1\). This results in an equation: \(y-4=(-3/4)(x+6)\). Simplifying this equation will give the equation of the parallel line.

Equation Perpendicular to the Given Line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. So, the slope of the line perpendicular to our given line would be the negative reciprocal of \(-3/4\), which is \(4/3\). Plugging this slope along with the given point \(-6,4\) into the slope-point form gives us: \(y-4=(4/3)(x+6)\). Simplifying this equation will give the equation of the perpendicular line.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when dealing with linear equations. The slope-intercept form is written as:
\[ y = mx + b \]

In this equation, \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is where the line crosses the y-axis. To convert a standard form equation, \( ax + by = c \), to slope-intercept form, solve for \( y \) as follows:

• Isolate the \( y \)-variable on one side of the equation.
• Manipulate the equation to solve for \( y \), resulting in the slope-intercept form.

For example, the given equation \( 3x + 4y = 7 \) can be transformed by subtracting \( 3x \) from both sides and then dividing by \( 4 \), arriving at \( y = -\frac{3}{4}x + \frac{7}{4} \), where the slope \( m \) is \( -\frac{3}{4} \) and the y-intercept \( b \) is \( \frac{7}{4} \).

Slope is a measure of the steepness of a line. A positive slope means the line rises as it moves left to right, while a negative slope means it falls. The absolute value of the slope determines the steepness, independent of direction.

To find the equation of a line, knowing only a point on the line and its slope, we use the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the line. This formula can then be simplified to the slope-intercept form for ease of graphing and analysis.

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart and have the same slope. To write the equation of a line that is parallel to another, we use the fact that parallel lines have identical slopes.

Given a point and a line, creating a new line parallel to the given one requires us to:

• Use the same slope as the given line.
• Use the point-slope form with the given point and the identified slope.

From our exercise, we use the point \( (-6,4) \) and the slope of the given line, which is \( -\frac{3}{4} \). By inserting these values into the point-slope form, we establish the equation \( y - 4 = -\frac{3}{4}(x + 6) \) of the line parallel to the given line. This equation can be simplified further to fit the slope-intercept form, which makes graphing straightforward.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). The slopes of two perpendicular lines are negative reciprocals of each other. That means if one line has a slope of \( m \), the line perpendicular to it will have a slope of \( -\frac{1}{m} \).

To find the equation of a line perpendicular to a given line and passing through a certain point, we:

• Calculate the negative reciprocal of the given line's slope.
• Apply the point-slope form using this new slope and the given point.

So, for our problem, the perpendicular slope to \( -\frac{3}{4} \) is \( \frac{4}{3} \). Using the point \( (-6,4) \) and our new slope in the point-slope form, we get \( y - 4 = \frac{4}{3}(x + 6) \), which is the equation of the line perpendicular to the given line. Like parallel lines, once we have the point-slope form, we can rearrange it into the slope-intercept form to make the relationship more apparent and to facilitate graphing and analysis.

It's worth noting that understanding the relationships between parallel and perpendicular slopes is crucial in coordinate geometry, especially in construction and design, where right angles are often required.