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Chapter 7: Problem 5

Graph the equations. $$ y=5 x-4 $$

Short Answer

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Answer: To graph the linear equation y = 5x - 4, follow these steps: 1. Identify the equation in slope-intercept form, which is y = mx + b. In this case, m (slope) = 5, and b (y-intercept) = -4. 2. Plot the y-intercept (-4) on the graph as the first point, which is (0, -4). 3. Find another point on the graph using the slope, in this case, 5/1. Starting from the y-intercept (0, -4), move 1 unit to the right and 5 units up to reach the point (1, 1). 4. Draw a straight line connecting the two points (0, -4) and (1, 1), and extend the line in both directions to represent the graph of the equation y = 5x - 4.

Step by step solution

01

Identify the Equation in Slope-Intercept Form

The given equation is already in slope-intercept form (y = mx + b). Here, m (slope) = 5, and b (y-intercept) = -4.
02

Plot the Y-Intercept

Now that we have the y-intercept (-4), we can plot the first point on the graph. The y-intercept is the point where the line crosses the y-axis. In this case, the point is (0, -4). Plot this point on the graph.
03

Find Another Point Using the Slope

The slope (m) represents the change in y over the change in x (rise over run). In this case, the slope is 5, which can be written as 5/1. This means for every 1 unit increase in x, y will increase by 5 units. Starting from the y-intercept (0, -4), move 1 unit to the right and 5 units up. This will give us a new point: (1, 1). Plot this point on the graph.
04

Draw the Line

Now that we have two points, (0, -4) and (1, 1), we can draw the line representing the equation y = 5x - 4. Connect the two points with a straight line, and extend the line beyond the points in both directions. This line represents the graph of the equation y = 5x - 4.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a particular way of setting up the terms of your equation, making graphing straightforward. It is expressed as \(y = mx + b\).
This form is useful because it directly reveals two crucial pieces of information:
  • **The slope** of the line, represented by \(m\), which tells us how steep the line is.
  • **The y-intercept** of the line, represented by \(b\), which shows where the line crosses the y-axis.
When an equation is written in this form, you can easily identify and plot these on a graph without additional calculations. You simply look at the equation, figure out \(m\) and \(b\), and you're ready to graph. This ease of use makes it the most common form for beginners when dealing with linear equations and graphing tasks.
Y-Intercept
The y-intercept \(b\) is the point where the line crosses the y-axis. It indicates the value of \(y\) when \(x\) is zero.

In the equation \(y = 5x - 4\), the y-intercept is -4. This tells us immediately that the line will intersect the y-axis at the point (0, -4).

When graphing, you start by plotting this point:
  • Place a dot at the y-intercept on the graph, which gives you a starting point.
  • Since it lies on the y-axis, you don鈥檛 need to worry about the x-value at this part of the graph, making it straightforward to identify.
This makes the y-intercept a critical component in understanding how the line sits on the coordinate plane.
Slope
The slope \(m\) of a line tells you how much the line rises or falls as you move horizontally along the x-axis. It is the 'ris\(e\) over run', or the change in \(y\) divided by the change in \(x\).
In \(y = 5x - 4\), the slope is 5, which can be translated to \(\frac{5}{1}\). This means that for every 1 unit you move to the right on the x-axis, move up 5 units on the y-axis.

Here's how to use the slope to graph a line:
  • From the y-intercept, use the slope to determine your next point. For example, from (0, -4), moving with the slope of \(5/1\), you would step to (1, 1).
  • The slope tells you the direction and steepness of the line, key in ensuring your line is accurately placed.
Understanding the slope gives you insights into the behavior of the line across the graph, whether it climbs up, slopes down, or is flat.

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Most popular questions from this chapter

We know that solutions to linear equations in two variables can be expressed as ordered pairs. Hence, the solutions can be represented as points in the plane. Consider the linear equation \(y=\) \(2 x-1\). Find at least ten solutions to this equation by choosing \(x\) -values between -4 and 5 and computing the corresponding \(y\) -values. Plot these solutions on the coordinate system below. Fill in the table to help you keep track of the ordered pairs.

Use the \(y\) -intercept and the slope to graph each line. $$ y=\frac{3}{4} x $$

Moving left to right, lines with _______ slope rise while lines with ________ slope decline.

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. $$ (0.00426,-0.00404),(-0.00191,-0.00404) $$

Graph the equations. $$ x=0 $$
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