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Chapter 4: Problem 91

In the following exercises, graph by plotting points. \(\frac{1}{3} x+y=2\)

Short Answer

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

Step by step solution

01

- Rewrite the equation in slope-intercept form

Start by isolating the variable y in the equation \(\frac{1}{3} x + y = 2\). Subtract \(\frac{1}{3} x\) from both sides to get \(y = -\frac{1}{3}x + 2\).
02

- Create a table of values

Choose several values for x and calculate the corresponding y-values using the equation \(y = -\frac{1}{3}x + 2\). For example:\* If \(x = 0 \Rightarrow y = 2\).* If \(x = 3 \Rightarrow y = 1\).* If \(x = -3 \Rightarrow y = 3\).
03

- Plot the points on a graph

Using a coordinate plane, plot the points obtained from the table (0, 2), (3, 1), and (-3, 3).
04

- Draw the line

Draw a straight line through the points plotted. This line represents the graph of the equation \(y = -\frac{1}{3}x + 2\).

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is crucial for graphing. The slope-intercept form is given by \(y = mx + b\), where:
  • m is the slope of the line.
  • b is the y-intercept, where the line crosses the y-axis.
For example, converting the equation \(\frac{1}{3} x + y = 2\) to slope-intercept form involves isolating y. We subtract \(\frac{1}{3} x\) from both sides to get \(y = -\frac{1}{3} x + 2\). In this form, the slope \(m = -\frac{1}{3}\) and the y-intercept \(b = 2\). This makes it easier to understand the behavior of the graph.
Coordinate Plane
The coordinate plane is a two-dimensional space defined by the x-axis (horizontal) and y-axis (vertical). Each point on the plane is represented by a pair of numbers \((x, y)\). The point \((0,0)\) is called the origin. When graphing equations, it's important to place each point accurately on this plane. For the equation \(y = -\frac{1}{3} x + 2\), we will plot points like \((0, 2)\), \((3, 1)\), and \((-3, 3)\). Each coordinate gives us a location on this plane where the line will pass through.
Plotting Points
Plotting points on a coordinate plane involves finding the specific (x, y) pairs that satisfy the equation.
  • First, choose a value for x.
  • Then, substitute it into the equation to find y.
This gives us a point \((x, y)\). For the equation \(y = -\frac{1}{3} x + 2\), let's choose x values: 0, 3, and -3. When we substitute these x values into the equation, we get:
  • If \(x = 0\), then \(y = 2\).
  • If \(x = 3\), then \(y = 1\).
  • If \(x = -3\), then \(y = 3\).
These points \((0, 2)\), \((3, 1)\), and \((-3, 3)\) are plotted on the coordinate plane.
Creating a Table of Values
A table of values is an organized way to list x-values along with their corresponding y-values. This helps in plotting points and drawing the graph of a linear equation. Here is how you create a table of values:
  • Pick several x-values (both positive and negative to see the line's behavior on both sides of the y-axis).
  • Use the equation to find the corresponding y-values for each x.
For instance, for the equation \(y = -\frac{1}{3} x + 2\), we can choose x-values like -3, 0, and 3. Calculating the y-values, we get a table:
  • \(x = -3\), \(y = 3\)
  • \(x = 0\), \(y = 2\)
  • \(x = 3\), \(y = 1\)
With these values, we can accurately plot the points on the coordinate plane and draw the line representing the equation.

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