Problem 29 - College Algebra with Corequisite Support

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: Find the Slope of the Given Line

The equation of the given line is \(3y = x - 4\). First, convert it to slope-intercept form \(y = mx + b\) by dividing all terms by 3, yielding \(y = \frac{1}{3}x - \frac{4}{3}\). The slope \(m\) of this line is \(\frac{1}{3}\).

Step 2: Determine the Slope of the Perpendicular Line

The slope of the line perpendicular to a given line is the negative reciprocal of the original slope. Given that the original slope is \(\frac{1}{3}\), the slope of the perpendicular line is \(-3\).

Step 3: Use the Point-Slope Formula

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

Step 4: Simplify to Slope-Intercept Form

Distribute the slope \(-3\) within the equation from Step 3: \[y - 1 = -3x - 6\]. Next, add 1 to both sides to solve for \(y\): \[y = -3x - 5\]. This is the slope-intercept form of the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that they are easy to understand and graph. This form is written as \(y = mx + b\).
Here, \(m\) is the slope of the line, which tells how steep the line is.
The \(b\) is the y-intercept, the point where the line crosses the y-axis. The slope-intercept form is popular because you can clearly see both the slope and the y-intercept just by looking at the equation.

If you want to go from another form, like the point-slope form \(y - y_1 = m(x - x_1)\), to the slope-intercept form, you rearrange the equation to solve for \(y\). This way, you isolate the y variable on one side of the equation.
For example, transforming \(3y = x - 4\) involved dividing by 3 to find \(y = \frac{1}{3}x - \frac{4}{3}\). The final equation, \(y = -3x - 5\), clearly shows that the slope is \(-3\) and the y-intercept is \(-5\). This format is very user-friendly for graphing and understanding line equations.

Perpendicular Lines

In geometry, perpendicular lines are lines that intersect at a right angle, which is 90 degrees. This angle is a key characteristic of perpendicular lines.
To know if two lines on a graph are perpendicular, their slopes will have a special relationship.
The slopes of perpendicular lines are negative reciprocals of each other. If you understand this relationship, you can easily check if two lines are perpendicular.

For example, consider a line with the equation \(y = \frac{1}{3}x - \frac{4}{3}\). The slope of this line is \(\frac{1}{3}\). Now, if we have another line perpendicular to it, its slope must be the negative reciprocal.
This means the slope will be \(-3\). Perpendicular lines are visually easy to spot because they form an "L" shape, but mathematically, it's all about that slope relationship!

Negative Reciprocal Slope

The concept of the negative reciprocal slope is all about finding how two lines relate to each other on a graph. Specifically, it helps in determining if lines are perpendicular. A slope tells us how slanted a line is, and a reciprocal is simply flipping a fraction.
So, if you take the slope of one line, its negative reciprocal is obtained by flipping the slope and changing its sign.

Let's break it down with an example: If one line has a slope of \(\frac{1}{3}\), the negative reciprocal of this slope would be \(-3\). You flip \(\frac{1}{3}\) to get \(3\) and then change the sign to negative.
This means any line with a slope of \(-3\) will be perpendicular to the line with a slope of \(\frac{1}{3}\).
Understanding how to find the negative reciprocal allows you to quickly determine perpendicularity in coordinate geometry.
This knowledge is crucial whenever you're dealing with properties of shapes, especially right angles.