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Chapter 4: Problem 3

Each of the following equations is in slope-intercept form. Identify the slope and the \(y\) -intercept, then graph each line using this information. $$y=\frac{2}{5} x-6$$

Short Answer

Expert verified
In the equation \(y = \frac{2}{5}x - 6\), the slope (m) is \(\frac{2}{5}\), and the y-intercept (b) is -6. To graph the line, plot the y-intercept (0, -6) and use the slope to find a second point: move 5 units to the right and 2 units up to get (5, -4). Draw a line through these two points.

Step by step solution

01

Identify the slope

To identify the slope, look for the coefficient of the x term in the given equation, which is in the slope-intercept form (y = mx + b). The coefficient of the x term is the slope (m). In this equation, \(y=\frac{2}{5} x-6\), the slope (m) is \(\frac{2}{5}\).
02

Identify the y-intercept

To identify the y-intercept, look for the constant term in the given equation, which is in the slope-intercept form (y = mx + b). The constant term is the y-intercept (b). In this equation,\( y=\frac{2}{5} x-6 \), the y-intercept (b) is -6.
03

Find two points on the line using slope and y-intercept

To graph the line, we need to find two points on the line. We already have the y-intercept as one point, which is (0, -6). Now, using the slope, we will find the second point. Since the slope (m) is \(\frac{2}{5}\), this means that for every 5 units moved to the right on the x-axis, the line goes up by 2 units on the y-axis. From the y-intercept (0, -6), move 5 units to the right on the x-axis and 2 units up on the y-axis. The second point will be at (5, -4).
04

Graph the line using the two points

Now that we have two points on the line, we can graph the line. Put the two points (0, -6) and (5, -4) on the graph. Then, draw a line that goes through these two points. This line represents the equation \(y=\frac{2}{5} x-6\).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in mathematics that involves representing equations on a coordinate plane. Linear equations can be easily graphed once they are in the slope-intercept form, which is given by the equation \(y = mx + b\). This form tells us exactly where to start (at the \(y\)-intercept, \(b\)) and how to proceed using the slope \(m\).
To graph a linear equation, such as \(y = \frac{2}{5}x - 6\), you need to identify two key components: the slope and the \(y\)-intercept. Once identified, these components facilitate the construction of the graph. Let's discuss the steps in detail.
Identifying Slope
The slope is a crucial part of the equation, often denoted by \(m\) in the slope-intercept form \(y = mx + b\). It signifies the steepness and direction of the line. In our example \(y = \frac{2}{5}x - 6\), the slope \(m\) is \(\frac{2}{5}\).
This means that for every 5 units you move horizontally to the right along the x-axis, the line rises 2 units vertically.
Understanding slope helps in predicting how the graph behaves.
  • If the slope is positive, like in this case, the line ascends as it moves from left to right.
  • A negative slope would mean the line descends.
  • If the slope is zero, the line is horizontal, indicating no vertical change.
  • An undefined slope corresponds to a vertical line.
Identifying Y-intercept
The \(y\)-intercept is where the line crosses the \(y\)-axis. It is represented by the constant term \(b\) in the slope-intercept equation \(y = mx + b\). For the equation \(y = \frac{2}{5}x - 6\), the \(y\)-intercept \(b\) is \(-6\).
This tells us that the line will intersect the \(y\)-axis at the point (0, -6).
The \(y\)-intercept is crucial because it provides a starting point for graphing the equation.
  • This point is always straightforward to identify and plot.
  • It provides a reference for applying the slope to find additional points on the line.
Plotting Points on a Graph
Once the \(y\)-intercept and slope have been identified, the next step is plotting these on a graph to visualize the line represented by the equation. Start by plotting the \(y\)-intercept, which is the point (0, -6) for our equation.
From this point, apply the slope \(\frac{2}{5}\) to find another point on the line. This means moving 5 units to the right and 2 units up, landing at the point (5, -4).
These two points can now be connected to form the line.
  • Use a ruler to ensure accuracy and extend the line across the grid.
  • Double-check the placement of points to avoid errors in slope representation.
  • Remember, the graph of a linear equation like this will always be a straight line.
By understanding and following these steps, you can efficiently graph any linear equation in slope-intercept form.

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Most popular questions from this chapter

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(y=0\) containing \(\left(-3,-\frac{5}{2}\right)\)

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{8}{2 x+1}$$

Graph each function using the slope and \(y\) -intercept. $$h(x)=-\frac{5}{2} x+4$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((0,2)\) and \((6,0)\)

Graph each function. $$A(r)=-3 r$$
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