Problem 29 - College Algebra

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. perpendicular to \(3 y=x-4\) and passes through the point (-2,1)

Step-by-Step Solution

Answer

The equation is \(y = -3x - 5\).

Step 1: Identify the slope of the given line

First, let's rewrite the given line equation \(3y = x - 4\) in the slope-intercept form \(y = mx + b\). Divide the entire equation by 3 to solve for \(y\): \[y = \frac{1}{3}x - \frac{4}{3}\]. Here, the slope \(m\) is \(\frac{1}{3}\).

Step 2: Find the perpendicular slope

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, take the negative reciprocal of \(\frac{1}{3}\), which is \(-3\). This is the slope of the line we're trying to find.

Step 3: Apply the point-slope formula

Use the point-slope formula \(y - y_1 = m(x - x_1)\), where \(m\) is the perpendicular slope \(-3\) and \((x_1, y_1)\) is the point \((-2, 1)\). Substitute the values: \[y - 1 = -3(x + 2)\].

Step 4: Simplify the equation into slope-intercept form

Expand the equation from Step 3: \[y - 1 = -3x - 6\]. Add 1 to both sides to solve for \(y\): \[y = -3x - 5\]. This is the equation of the line in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a line. It's generally written as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The slope \( m \) indicates how much \( y \) changes for a unit change in \( x \). When you know the slope and the y-intercept, you can easily graph the line by plotting the y-intercept and using the slope to find another point on the line. For instance, in the exercise above, the final line equation \( y = -3x - 5 \) tells us that the line crosses the y-axis at -5 and has a slope of -3.

Perpendicular Slope

Understanding perpendicular slopes is crucial when working with geometric concepts involving lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. For example, if the original line has a slope \( m \), a line perpendicular to it will have a slope of \(- \frac{1}{m} \). This reciprocal relationship creates a 90-degree angle between the two lines.
In the exercise, the original line has a slope of \( \frac{1}{3} \). Therefore, the perpendicular slope is \(-3\), since \(-3\) is the negative reciprocal of \( \frac{1}{3} \). Being familiar with this concept can greatly help when required to create or interpret geometric arrangements involving perpendicular lines.

Line Equation

A line equation is a mathematical description of a straight line on a coordinate plane. It is used to determine the position of points that lie on the line. When working with line equations, several forms can be used, such as the slope-intercept form and point-slope form.
In the exercise, we began with a point-slope form \( y - y_1 = m(x - x_1) \), which helps to express a line when a point on the line and the slope are known. Using this form, you can plug in the coordinates \((x_1, y_1)\) of a known point, and the slope \(m\) to find the equation.
For example, with a slope \(-3\) and through the point \((-2,1)\), you substitute these into the formula:

• \( y - 1 = -3(x + 2) \)

This equation, when simplified, results in the familiar slope-intercept form: \( y = -3x - 5 \). Understanding how to move between different forms of line equations is crucial for solving many algebra problems.