Problem 76 - Calculus Volume 1

Question

For the following exercises, 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

The slope is \(-\frac{1}{7}\), the y-intercept is 1, and the line crosses at points (0,1) and (7,0).

Step 1: Identify the form of the linear equation

The equation given is in the slope-intercept form, which is written as \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Extract the slope

From the equation \( y = -\frac{1}{7}x + 1 \), identify the slope \( m \) as \( -\frac{1}{7} \).

Step 3: Identify the y-intercept

In the equation \( y = -\frac{1}{7}x + 1 \), the constant term is \( 1 \), which is the y-intercept \( b \). Thus, the y-intercept is \( b = 1 \).

Step 4: Plot the y-intercept

On a graph, plot the point \( (0,1) \), since the y-intercept \( b \) indicates where the line crosses the y-axis.

Step 5: Use the slope to find another point

Starting from the y-intercept \( (0,1) \), use the slope \( -\frac{1}{7} \), which means for every 7 units you move to the right on the x-axis, move 1 unit down on the y-axis. Plot the point \( (7, 0) \).

Step 6: Draw the line

Draw a straight line through the points \( (0,1) \) and \( (7,0) \). Extend the line across the graph, ensuring it passes through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a handy way to express linear equations. It's written as \( y = mx + b \), where:

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

This form makes it easy to quickly identify the slope and y-intercept, which are key to understanding and graphing a linear equation. The slope-intercept form is very popular because it straightforwardly shows how the y-value changes with the x-value, and you can also see right away where the line hits the y-axis.

Graphing Linear Equations

Graphing linear equations is all about finding points that lie on the line described by the equation and connecting them. Start by plotting the y-intercept, easily found using the slope-intercept form equation. In our example, the y-intercept is \( (0, 1) \).

• Locate the y-intercept on the graph and plot it.
• Use the slope to find another point. The slope tells you how to move from the y-intercept to plot more points.

Once you have two points, like \( (0,1) \) and another from using the slope, draw a line through these points. Extend it in both directions. This line represents all solutions to the equation.

Slope

The slope of a line expresses its steepness and direction. It is found in the equation \( y = mx + b \) as \( m \). Simply put, the slope is the rate of change of y with respect to x. For example:

• If \( m > 0 \), the line rises as it moves from left to right.
• If \( m < 0 \), the line falls as it moves from left to right.
• If \( m = 0 \), the line is perfectly horizontal, indicating no change in y.

In our equation \( y = -\frac{1}{7}x + 1 \), the slope \( m \) is \(-\frac{1}{7}\), meaning the line falls slightly as you move to the right. You move down 1 unit for every 7 units you go to the right.

Y-Intercept

The y-intercept is a particularly crucial part of a linear equation in slope-intercept form. Denoted by \( b \) in \( y = mx + b \), it shows where the line crosses the y-axis. This point is easy to find, as it is the value of \( y \) when \( x \) equals zero.

• In our example, the y-intercept is 1, meaning the line passes through the point \( (0, 1) \).

Identifying the y-intercept allows you to graph the line efficiently, provided you also know the slope. It presents a clear starting point from which you can use the slope to find other points on the line.