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Chapter 3: Problem 40

Graph each linear equation. Plot four points for each line. $$y=-\frac{2}{3} x$$

Short Answer

Expert verified
Plot the points (-3, 2), (-1, \(\frac{2}{3}\)), (1, -\(\frac{2}{3}\)), and (3, -2) and draw the line through them.

Step by step solution

01

Understanding the equation

The equation given is in the slope-intercept form, which is generally written as \(y = mx + b\). Here, \(m = -\frac{2}{3}\) (the slope) and \(b = 0\) (the y-intercept). This means the line passes through the origin (0, 0).
02

Choose values for x

To plot the line, choose four different values for \(x\). Let's use \(-3, -1, 1, 3\).
03

Calculate corresponding y values

Using the equation \(y = -\frac{2}{3} x\), calculate the y values for each chosen x value:- For \(x = -3\): \(y = -\frac{2}{3}(-3) = 2\)- For \(x = -1\): \(y = -\frac{2}{3}(-1) = \frac{2}{3}\)- For \(x = 1\): \(y = -\frac{2}{3}(1) = -\frac{2}{3}\)- For \(x = 3\): \(y = -\frac{2}{3}(3) = -2\)
04

Plot the points

On a coordinate plane, plot the points calculated above: (-3, 2), (-1, \(\frac{2}{3}\)), (1, -\(\frac{2}{3}\)), and (3, -2).
05

Draw the line

Connect the points with a straight line. This represents the graph of the linear equation \(y = -\frac{2}{3} x\).

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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 main ways to represent a linear equation is through the slope-intercept form, written as \(y = mx + b\). Here, \(m\) represents the slope of the line, which shows how steep the line is, and \(b\) is the y-intercept, indicating the point where the line crosses the y-axis. For our exercise, \(y = -\frac{2}{3} x\), the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is 0. This tells us that our line descends from left to right and crosses the y-axis at the origin (0, 0).
Coordinate Plane
A coordinate plane is a two-dimensional surface defined by two number lines intersecting at right angles. These number lines are called axes. The horizontal number line is known as the x-axis and the vertical one is called the y-axis. The point where these two axes intersect is called the origin and has coordinates (0, 0). To plot points, we use pairs of numbers (x, y), known as coordinates, which indicate positions on the plane relative to these axes.
Plotting Points
Plotting points involves locating positions on the coordinate plane based on their coordinates (x, y). For our linear equation \(y = -\frac{2}{3} x\), we calculated several points to graph. For example, when \(x = -3\), \(y = 2\). This point is plotted by finding -3 on the x-axis and 2 on the y-axis, then marking where these values intersect. Repeat this for the other x values and their corresponding y values. The points we plotted are (-3, 2), (-1, \(\frac{2}{3}\)), (1, -\(\frac{2}{3}\)), and (3, -2).
Linear Equations
Linear equations form straight lines when graphed on a coordinate plane. These equations typically depict a constant rate of change, which is represented by the slope in the slope-intercept form. In our exercise, this constant rate is \(-\frac{2}{3}\), meaning for every 3 units we move to the right on the x-axis, we move 2 units down on the y-axis. Graphing the points (-3, 2), (-1, \(\frac{2}{3}\)), (1, -\(\frac{2}{3}\)), and (3, -2) and connecting them with a straight line visualizes the equation \(y = -\frac{2}{3} x\). This is how linear equations are visually represented.

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