Problem 47 - Algebra for College Students

Question

In each case find the slope of line I and graph both lines that are mentioned. Line \(l\) is perpendicular to a line with slope \(\frac{4}{5}\). Both lines contain the origin.

Step-by-Step Solution

Answer

The slope of line \(l\) is \(-\frac{5}{4}\). Both lines can be graphed through the origin with their respective slopes.

Step 1 - Identify the Given Slope

The given slope of the line we are dealing with is \(\frac{4}{5}\). This information will be used to find the slope of the perpendicular line.

Step 2 - Recall the Property of Perpendicular Slopes

When two lines are perpendicular, the product of their slopes is -1. This means if one line has a slope of \(m\), the perpendicular line will have a slope of \(-\frac{1}{m}\).

Step 3 - Calculate the Slope of the Perpendicular Line

Given the slope \(\frac{4}{5}\), the slope of the line perpendicular to it is: \(-\frac{1}{\frac{4}{5}} = -\frac{5}{4}\). Thus, the slope of line \(l\) is \(-\frac{5}{4}\).

Step 4 - Graph the Lines

Graph both lines using their slopes and the fact that they pass through the origin (0,0). The first line with slope \(\frac{4}{5}\) passes through (0,0). Plot a point at (5, 4) and draw the line through these points. The perpendicular line has a slope of \(-\frac{5}{4}\). Plot a point at (4, -5) and draw the line through (0,0) and (4,-5).

Key Concepts

slope of a line

To understand any linear equation, you need to grasp the concept of the slope of a line. The slope tells us how steep the line is. Mathematically, the slope is defined as the ratio of the 'rise' (change in the y-coordinate) to the 'run' (change in the x-coordinate). If you have two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), you can calculate the slope using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Consider a line passing through points \( (3, 2) \) and \( (8, 5) \). You would compute the slope like this: \( m = \frac{5 - 2}{8 - 3} = \frac{3}{5} \).

A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. Horizontal lines have a slope of 0, and vertical lines have an undefined slope.

perpendicular slopes

Perpendicular lines intersect to form a right angle (90 degrees). To determine if two lines are perpendicular, you need to understand their slopes. If the slope of one line is \( m \), then the slope of a line perpendicular to it will be \( -\frac{1}{m} \). This relationship exists because the product of the slopes of two perpendicular lines is always \(-1\).

For example, if a line has a slope of \( \frac{4}{5} \), a line perpendicular to it will have a slope of \( - \frac{5}{4} \). This can be shown with the equation \( \frac{4}{5} \times - \frac{5}{4} = -1 \).

Knowing this is crucial when graphing or working with linear equations because it helps you quickly identify perpendicular lines.

graphing linear equations

Graphing linear equations involves plotting points on a coordinate plane and drawing the line that connects them. Each linear equation can be written in the slope-intercept form \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

To graph a line using its slope and y-intercept:

• Start at the y-intercept \( (0, b) \) on the y-axis.
• Use the slope to determine the next points (rise/run). For a slope of \( \frac{4}{5} \), move 4 units up and 5 units right.
• Plot these points and draw a line through them.

For perpendicular lines, plot the y-intercepts first. Then, use the negative reciprocal of the first line's slope to determine the points for the second line. For example, a line with slope \( \frac{4}{5} \) can be plotted through the origin (0,0) and another point like (5,4). Its perpendicular line with slope \( -\frac{5}{4} \) can go through the origin and (4, -5).

Graphing these lines lets you visualize their relationships easily and understand their geometric properties.