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Chapter 3: Problem 38

Graph each linear equation using the slope and y-intercept. $$y=-\frac{4}{3} x$$

Short Answer

Expert verified
The graph of the equation \(y = -\frac{4}{3}x\) is a straight line through the origin with slope -4/3, moving four steps down for every three steps to the right.

Step by step solution

01

Identify the y-intercept

The equation \(y= -\frac{4}{3}x\) is the same as \(y=-\frac{4}{3}x+0\). Thus, the y-intercept (where the line crosses the y-axis) is at the origin (0,0).
02

Understand the Slope

The slope of this line is -4/3. This means for each three units you go to the right on x-axis, you go four units down on the y-axis, since it's negative.
03

Graph the Line

Plot the y-intercept at (0,0). From there, follow the slope by moving three units to the right and four units down repeatedly until a line can be drawn through the plotted points, clearly representing the linear equation.

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Key Concepts

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

Slope
The slope is a key feature of linear equations, expressed as the ratio of how much the line goes up or down for each step it goes right. It tells us how steep the line is. In the equation \(y = -\frac{4}{3}x\), the slope is \(-\frac{4}{3}\). This means:
  • For every 3 units you move to the right, the line drops 4 units.
  • The negative sign indicates the line goes downward as you move from left to right.
Using the slope, you can easily plot points along the line once you know where it starts. Remember, the slope is often represented as \(\frac{rise}{run}\), where 'rise' is how much you go up or down, and 'run' is how much you move left or right. In this case, it鈥檚 a fall of 4 (negative rise) per 3 right units of run. This information is crucial for correctly graphing the line.
y-intercept
The y-intercept is where the line crosses the y-axis. It shows the value of \(y\) when \(x = 0\). In our equation \(y = -\frac{4}{3}x + 0\), the y-intercept is 0. This means the line crosses the y-axis at the point (0,0).
The y-intercept provides a starting point on the graph, which is essential for drawing the line:
  • Begin by plotting the y-intercept.
  • Use this point as the anchor to apply the slope.
By understanding the y-intercept, you can more accurately represent linear equations on a graph. It simplifies the graphing process, as you always know where to begin drawing the line.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph equations. It consists of two axes:
  • The horizontal line is the x-axis.
  • The vertical line is the y-axis.
These axes split the plane into four quadrants. Each point on this plane is identified by a pair of numbers (x, y), called coordinates.
To graph a linear equation like \(y = -\frac{4}{3}x\), follow these steps:
  • Start at the y-intercept.
  • Move according to the slope.
Understanding the coordinate plane helps in visualizing how the slope and y-intercept interact. It's a foundational tool for graphing any linear equation and is used to identify the placement and direction of lines. Mastering the coordinate plane allows you to see math come to life visually.

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Most popular questions from this chapter

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The graphs of \(2 x-3 y=-18\) and \(-2 x+3 y=18\) must have the same intercepts because I can see that the equations are equivalent.

A nursery offers a package of three small orange trees and four small grapefruit trees for \(\$ 22\). a. If \(x\) represents the cost of one orange tree and \(y\) represents the cost of one grapefruit tree, write an equation in two variables that reflects the given conditions. b. If a grapefruit tree costs \(\$ 2.50,\) find the cost of an orange tree.

What is the graph of an equation?

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\)-intercept and the \(y\)-intercept. \(2 x+y=4\)
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