Problem 28 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in the form \(x=a, y=b\), or \(y=m x+b\). See Examples 5 and \(6 .\) Through (-4,8)\(;\) perpendicular to \(2 x-3 y=1\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{3}{2}x + 2\).

Step 1: Identify the Slope of the Given Line

The given line equation is \(2x - 3y = 1\). First, convert this equation into slope-intercept form \(y = mx + b\) to find its slope. Rearrange the equation: \(-3y = -2x + 1\) and then divide by -3 to get \(y = \frac{2}{3}x - \frac{1}{3}\). Now, we see that the slope (\(m\)) of the given line is \(\frac{2}{3}\).

Step 2: Find the Slope of the Perpendicular Line

Lines that are perpendicular have slopes that are negative reciprocals of each other. Thus, if the slope of the given line is \(\frac{2}{3}\), the slope of the line perpendicular to it will be \(-\frac{3}{2}\).

Step 3: Use the Point-Slope Form

We have the slope of the desired line as \(-\frac{3}{2}\) and it passes through the point (-4, 8). Use the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point. Substitute the known values: \(y - 8 = -\frac{3}{2}(x + 4)\).

Step 4: Simplify to Slope-Intercept Form

Expand and simplify the equation from Step 3: \(y - 8 = -\frac{3}{2}x - 6\). Add 8 to each side of the equation to solve for \(y\): \(y = -\frac{3}{2}x + 2\). Now we have the equation of the line in the desired form \(y = mx + b\).

Key Concepts

Slope-Intercept Form

When working with linear equations, the slope-intercept form is one of the most commonly used formats. It provides a straightforward way to write equations of lines and understand their slope and y-intercept with ease. The general expression is given as:
\[ y = mx + b \]
Here:

• \(m\) represents the slope of the line. The slope indicates how steep a line is or its rate of change.
• \(b\) is the y-intercept. It's the point where the line crosses the y-axis ((x=0)).

Using this format, you can quickly visualize the behavior of a line on a graph. For example, in our exercise we converted the line given by \(2x - 3y = 1\) into the slope-intercept form \(y = \frac{2}{3}x - \frac{1}{3}\). This tells us that the slope is \(\frac{2}{3}\), and the line crosses the y-axis at \(-\frac{1}{3}\). By understanding these two components, predicting the trajectory of the line becomes intuitive.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which is an angle of 90 degrees. Understanding the relationship between their slopes is key. When two lines are perpendicular, their slopes are negative reciprocals of each other. If the slope of one line is \(a/b\), then the slope of the line perpendicular to it will be \(-b/a\).
This characteristic helps in determining the slope of a perpendicular line without needing to graph it first. Let's say we have a line with a slope of \(\frac{2}{3}\), then a line perpendicular to it would necessarily have a slope of \(-\frac{3}{2}\).
In our problem, we needed a line that is perpendicular to \(2x - 3y = 1\). As calculated, the perpendicular slope turns out to be \(-\frac{3}{2}\). Thus, calculating the negative reciprocal is a handy tool when dealing with problems involving perpendicular lines.

Point-Slope Form

The point-slope form of a line is extremely useful when you know a point through which the line passes and its slope. The general formula is:
\[y - y_1 = m(x - x_1) \]
In this formula:

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

This form is convenient because it allows you to write the equation of a line without needing to immediately rearrange it into slope-intercept form. In our problem, we used point \((-4, 8)\) and slope \(-\frac{3}{2}\) to get the equation:
\[y - 8 = -\frac{3}{2}(x + 4) \]
Furthermore, point-slope form is flexible and when expanded, can be effortlessly converted into the slope-intercept format, making it a versatile tool in algebra.