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Chapter 11: Problem 62

In the following exercises, graph by plotting points. $$ y=-\frac{5}{3} x+4 $$

Short Answer

Expert verified
Plot (0, 4) and (3, -1). Draw a line through these points.

Step by step solution

01

- Understand the Equation

The equation given is in the slope-intercept form of a line: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = -\frac{5}{3} \) and \( b = 4 \).
02

- Identify the Y-Intercept

The y-intercept is where the graph crosses the y-axis. In this case, it is 4. So, plot the point (0, 4) on the graph.
03

- Determine the Slope

The slope of the line is \( -\frac{5}{3} \). This means that for every 3 units you move to the right, you move 5 units down. Use this information to find another point on the line.
04

- Plot a Second Point Using the Slope

Starting from the point (0, 4), move 3 units to the right (positive x-direction) and 5 units down (negative y-direction). This gives you the point (3, -1). Plot this point on the graph.
05

- Draw the Line

With the points (0, 4) and (3, -1) plotted, draw a straight line through these points. This line is the graph of the equation \( y = -\frac{5}{3} x + 4 \).

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Key Concepts

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

slope-intercept form
To understand how to graph a linear equation, it's crucial to recognize the slope-intercept form. This form is written as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. In this format, the equation is easy to read and interpret. You can quickly identify the starting point of the graph (the y-intercept) and how steep the line is (the slope). This form is very versatile and widely used.
plotting points
Plotting points is a fundamental step in graphing any equation. First, identify the points you need to plot by using values of \( x \) from the equation. For example, if we start with the equation \( y = -\frac{5}{3}x + 4 \), we can find points by substituting different values of \( x \) and solving for \( y \). Begin by plotting the y-intercept, which in this case is the point (0, 4). After that, use the slope to find another point. By continuing this process, you build a series of points that, when connected, form the graph.
determining slope
The slope of a line indicates its steepness and direction. It is calculated as the rise over run, meaning the change in \( y \) divided by the change in \( x \). In the equation \( y = -\frac{5}{3}x + 4 \), the slope \( m = -\frac{5}{3} \). This tells us that for every 3 units we move to the right along the x-axis, we move 5 units down along the y-axis. Knowing this allows you to find another point on the graph by starting from a known point and applying the rise and run values according to the slope.
y-intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept. For our equation \( y = -\frac{5}{3}x + 4 \), the y-intercept is 4. This means that the line crosses the y-axis at (0, 4). Plotting this point is always the first step when graphing a line from the slope-intercept form equation. From there, the slope is used to find additional points to draw the line.

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Most popular questions from this chapter

In the following exercises, find three solutions to each linear equation. $$ y=-x-1 $$

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (-1,-1),(0,-5) $$

In the following exercises, graph each equation. $$ x-y=3 $$

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (2,1),(4,5) $$

In the following exercises, graph by plotting points. $$ 2 x+3 y=12 $$
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