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Chapter 3: Problem 114

In the following exercises, graph the line of each equation using its slope and \(y\) -intercept. $$ y=-\frac{3}{5} x+2 $$

Short Answer

Expert verified
Slope: \(-\frac{3}{5}\), Y-intercept: 2. Plot (0, 2) and (5, -1); draw the line.

Step by step solution

01

Identify the Slope and Y-intercept

The given equation is in the slope-intercept form, which is written as \( y = mx + b \). Here, \( m \) (the coefficient of \( x \) ) is the slope and \( b \) is the y-intercept. For the equation \( y = -\frac{3}{5}x + 2 \) , the slope \( m \) is \( -\frac{3}{5} \) and the y-intercept \( b \) is 2.
02

Plot the Y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For \( b = 2 \), plot the point (0, 2) on the graph.
03

Use the Slope to Find Another Point

From the y-intercept (0, 2), use the slope to find another point on the graph. The slope of \( -\frac{3}{5} \) means that for every 5 units you move to the right (positive direction on x-axis), you move 3 units down (negative direction on y-axis). Starting from (0, 2), move 5 units to the right to (5, 2) and then move 3 units down to (5, -1). Plot the point (5, -1).
04

Draw the Line

Draw a straight line through the two points (0, 2) and (5, -1). Extend the line in both directions, and you have the graph of the equation \( y = -\frac{3}{5}x + 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
One of the most common ways to represent linear equations is by using the slope-intercept form. It is written as: \( y = mx + b \) , where:
  • \( y \) represents the dependent variable (usually on the vertical axis).
  • \( x \) represents the independent variable (usually on the horizontal axis).
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept (the point where the line crosses the y-axis).
The slope-intercept form provides a straightforward way to understand the behavior of linear equations and makes it easy to graph them.
slope
The slope of a line indicates its steepness and direction. It is represented by the letter \( m \) in the slope-intercept form equation. The formula for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In the equation \( y = -\frac{3}{5}x + 2 \) , the slope is \( -\frac{3}{5} \). This means that for every 5 units you move to the right along the x-axis, you move 3 units down on the y-axis. The slope can be positive, negative, zero, or undefined:
  • A positive slope means the line goes upward as you move to the right.
  • A negative slope means the line goes downward as you move to the right.
  • A zero slope means the line is horizontal.
  • An undefined slope means the line is vertical.
Understanding the slope is crucial for graphing linear equations and understanding the line's behavior.
y-intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). To locate the y-intercept, simply look at the constant term in the equation. For instance, in the equation \( y = -\frac{3}{5}x + 2 \) , the y-intercept is 2. Therefore, the line crosses the y-axis at the point (0, 2). This point is essential when graphing the equation because it provides a starting point to apply the slope and find other points on the line. Knowing the intercept helps you quickly plot the line on a graph and understand how the line positions itself relative to the origin (0,0).
plotting points
Plotting points is a fundamental skill for graphing linear equations. Once you have the slope and the y-intercept, you can plot the initial point and use the slope to determine another. Follow these steps:
  • Identify the y-intercept and plot it on the y-axis. For \( y = -\frac{3}{5}x + 2 \), the y-intercept is 2, so you plot (0,2).
  • Use the slope to find another point. The slope \( -\frac{3}{5} \) tells you that for every 5 units you move right, you move down by 3 units. Starting from (0,2), move right to (5,2) and then down to (5,-1). Plot this point.
  • Draw a line through the points you鈥檝e plotted. Extend the line in both directions to complete the graph.
By learning to plot points based on the slope and y-intercept, you can easily graph any linear equation. This method breaks down the process into clear, manageable steps.

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