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Chapter 3: Problem 44

Find the slope of the line determined by each equation. $$y=-\frac{2}{3} x+6$$

Short Answer

Expert verified
The slope of the line determined by the equation \(y=-\frac{2}{3}x+6\) is \(-\frac{2}{3}\).

Step by step solution

01

Recognize the Slope-Intercept Form

The given equation is in the slope-intercept form. Here, \(y = -\frac{2}{3}x + 6\) which can be written in a general form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
02

Identify the Slope

The slope of the line is the coefficient of \(x\). In this case, it is \(-\frac{2}{3}\). This is the slope of the line represented by the given equation.

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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 is a way to express the equation of a straight line. It is commonly written as \( y = mx + b \),where:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, where the line crosses the y-axis.
This form is incredibly useful when you want to quickly identify the slope and the y-intercept from an equation. For example, in the equation from the exercise, \( y = -\frac{2}{3}x + 6 \), we can see right away that the slope \(m = -\frac{2}{3}\) and the y-intercept is\(b = 6\). This simplicity makes it easier to understand and graph linear equations, especially for beginners.
Linear Equations
Linear equations are equations that make straight lines when graphed on a coordinate plane. They take the form \( Ax + By = C \), where \(A\), \(B\), and \(C\) are constants.

The beauty of these equations lies in their predictability and simplicity. If the equation can be solved for \(y\), it can often be rewritten in the slope-intercept form \( y = mx + b \), which is handy for quickly identifying important characteristics like the slope and y-intercept.
  • Linear equations will always graph as a straight line.
  • The slope \(m\) tells us how steep the line is and the direction in which it slants.
  • If \(m\) is positive, the line goes upward as it moves from left to right. If \(m\) is negative, it descends.
In our example equation \( y = -\frac{2}{3}x + 6 \), the negative slope indicates a downward slant and its value tells us how steep that slant is.
Graphing Linear Functions
Graphing linear functions involves plotting their equations on a coordinate grid. This visualization makes understanding the relationship between variables much easier.

Here is how you would graph a linear function like\( y = -\frac{2}{3}x + 6 \):
  • Start at the y-intercept: This is the point where the line crosses the y-axis. For our equation, it is \( (0, 6) \).
  • Apply the slope: The slope \(-\frac{2}{3}\) tells us to go down by 2 units and right by 3 units from the y-intercept.
    • From point \( (0, 6) \), move down to \( (3, 4) \) and plot this point.
    • Keep repeating this if necessary to ensure your line is straight.
Once you have your points, simply draw a straight line through them.
This method helps you visually confirm that your math is correct and gives you a picture of how one variable changes in relation to the other. Graphing is a key skill in deciphering linear equations and matching visual data with its numerical counterpart.

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Most popular questions from this chapter

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x z\) represents _____ variation.

When graphing an equation in two variables, how many solutions of the equation must be found?

Explain the meaning of combined variation.

Graph each equation using any method. $$y=-3 x-1$$

Graph each equation using any method. $$x-y=-2$$
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