Problem 76 - Calculus Volume 1

Question

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. \(y=-\frac{1}{7} x+1\)

Step-by-Step Solution

Answer

Slope = \(-\frac{1}{7}\), \(y\)-intercept = 1. Graph starts at (0, 1) with a slope of \(-\frac{1}{7}\).

Step 1: Identify the Slope

The given linear equation is in the slope-intercept form, which is generally written as \(y = mx + b\). In this form, \(m\) is the slope. From the equation \(y = -\frac{1}{7}x + 1\), we can see that the slope \(m = -\frac{1}{7}\).

Step 2: Identify the Y-Intercept

Continuing with the slope-intercept form \(y = mx + b\), the \(b\) represents the \(y\)-intercept. From the equation \(y = -\frac{1}{7}x + 1\), we identify the \(y\)-intercept \(b = 1\). This means the line crosses the \(y\)-axis at the point \((0, 1)\).

Step 3: Graph the Line

To graph the line, start by plotting the \(y\)-intercept, the point \((0, 1)\) on the coordinate plane. From this point, use the slope \(m = -\frac{1}{7}\), which means for every 7 units you move to the right (positive direction on the x-axis), you move 1 unit down (negative direction on the y-axis). Plot another point using this slope. Connect the points with a straight line extending in both directions.

Key Concepts

Slope-Intercept Form

In mathematics, the slope-intercept form is a way to write linear equations. It is one of the most useful forms for graphing linear equations quickly. The general formula for the slope-intercept form is given by

• \( y = mx + b \)

where:

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

The slope-intercept form is particularly helpful because it immediately provides the slope and y-intercept of a line, making it easy to sketch the graph of the equation. By understanding this form, students can quickly interpret and graph lines, making it a critical skill in algebra.

Graphing Linear Equations

Graphing linear equations involves plotting them on a coordinate plane, which consists of an x-axis (horizontal) and y-axis (vertical). To graph a linear equation in slope-intercept form, start by marking the y-intercept on the y-axis.
The next step is to use the slope to find another point on the line. The slope is expressed as a fraction \( \frac{rise}{run} \). For example, a slope of \(-\frac{1}{7}\) indicates that for every 7 units you move to the right on the x-axis, you move 1 unit down on the y-axis.
Once you have two points plotted using the slope and y-intercept, draw a straight line through these points. This line is the graphical representation of the linear equation, stretching infinitely in both directions unless stated otherwise in a specific range.

Identifying Slope and Y-Intercept

Identifying the slope and y-intercept is important for understanding the behavior of a linear equation. The slope \( m \) indicates how steep the line is and in which direction it tilts.

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope indicates a horizontal line.

The y-intercept \( b \) is easy to find in the slope-intercept form, as it's simply the constant term added to the expression. It represents the starting point of the line on the y-axis.
Practicing with identifying these characteristics in the slope-intercept form allows students to quickly analyze and understand the linear equations, enabling them to efficiently solve algebra problems and predict the graph's behavior.