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Chapter 3: Problem 18

In the following exercises, graph by plotting points. $$ y=\frac{1}{3} x-1 $$

Short Answer

Expert verified
Plot points (-3, -2), (0, -1), and (3, 0) and draw a line through them.

Step by step solution

01

- Understand the Equation

The equation given is in the form of a linear equation: \[ y = \frac{1}{3}x - 1 \] This represents a straight line where the slope is \(\frac{1}{3}\) and the y-intercept is \(-1\).
02

- Choose x-values to Plot

Choose a few values for \( x \) to find the corresponding \( y \) coordinates. For simplicity, choose integer values like -3, 0, and 3.
03

- Calculate y-values

Substitute the chosen \( x \)-values into the equation to find \( y \): When \( x = -3 \): \[ y = \frac{1}{3}(-3) - 1 = -1 - 1 = -2 \] When \( x = 0 \): \[ y = \frac{1}{3}(0) - 1 = -1 \] When \( x = 3 \): \[ y = \frac{1}{3}(3) - 1 = 1 - 1 = 0 \]
04

- Plot the Points

Plot the calculated (x, y) points on the graph: ( -3, -2 ), ( 0, -1 ), and ( 3, 0 ).
05

- Draw the Line

Draw a straight line passing through all the plotted points. This will be the graph of the given linear equation.

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Key Concepts

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

Linear Equations
Linear equations are a fundamental concept in algebra. They create a straight line when graphed on a coordinate plane. A linear equation usually looks like this: \( y = mx + b \)Where:
  • \( y \): the dependent variable
  • \( x \): the independent variable
  • \( m \): the slope
  • \( b \): the y-intercept
For example, the linear equation \( y = \frac{1}{3}x - 1 \) represents a straight line with specific characteristics like slope and y-intercept.
Slope
The slope of a line shows how steep the line is. It's represented by \( m \) in the equation \( y = mx + b \). The slope tells us how much \( y \) changes for a certain change in \( x \). In our equation \( y = \frac{1}{3}x - 1 \), the slope is \( \frac{1}{3} \). This means that for every one unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \) units. Understanding slope is useful for predicting how changes in one variable affect the other.
Y-Intercept
The y-intercept is where the line crosses the y-axis, represented by \( b \) in \( y = mx + b \). In our example, \( b = -1 \). This means the line crosses the y-axis at \( -1 \). To find the y-intercept, set \( x \) to 0 in the equation:\( y = \frac{1}{3}(0) - 1 = -1 \)So when \( x \) is 0, \( y \) is \( -1 \). This is an important point for graphing the line as it gives a starting point for plotting.
Plotting Points
Plotting points involves finding specific points on the graph to draw the line. Here's how it is done:
  • Choose values for \( x \)
  • Calculate corresponding \( y \) values using the equation
  • Plot these (x, y) points on the graph
For example:
  • When \( x = -3 \): \( y = \frac{1}{3}(-3) - 1 = -2 \)
  • When \( x = 0 \): \( y = \frac{1}{3}(0) - 1 = -1 \)
  • When \( x = 3 \): \( y = \frac{1}{3}(3) - 1 = 0 \)
Plot these points on the graph: (-3, -2), (0, -1), and (3, 0). Finally, draw a line through these points to represent the linear equation.

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