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Chapter 5: Problem 32

Sketch the graph of the given equation. Label the intercepts. $$y=3 x$$

Short Answer

Expert verified
The graph of y = 3x is a line passing through the origin (0, 0) with a slope of 3.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in the form of y = mx + c, where m is the slope, and c is the y-intercept. Here, y = 3x, so the slope m = 3 and the y-intercept c = 0.
02

Determine the Intercepts

To find the intercepts, first find the y-intercept: Setting x = 0 in the equation y = 3x gives y = 0. Therefore, the y-intercept is (0, 0). To find the x-intercept, set y = 0, which also gives x = 0. Thus, the x-intercept is (0, 0).
03

Plot the Intercepts

On a coordinate plane, plot the intercepts found in the previous step. The point (0, 0) should be marked.
04

Plot Additional Points

To accurately sketch the line, select another value for x. For example, if x = 1, then y = 3(1) = 3, so the point (1, 3) can also be plotted. Similarly, if x = -1, then y = 3(-1) = -3, plot the point (-1, -3).
05

Draw the Line

Connect the points (0, 0), (1, 3), and (-1, -3) with a straight line. This line represents the graph of the equation y = 3x.
06

Label the Intercepts

Clearly label the intercept at (0, 0) on the graph to show where the line crosses the axes.

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Key Concepts

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

Coordinate Plane
A coordinate plane is like a map. It helps you locate points using two numbers: the x-coordinate and the y-coordinate. The plane is divided into four quadrants by a horizontal line called the x-axis and a vertical line called the y-axis. The point where these two axes meet is called the origin, represented as (0, 0).

To find any point on this plane, you look at how far it is from the origin in the horizontal direction (x-coordinate) and the vertical direction (y-coordinate). For example, the point (2, 3) is 2 units to the right of the origin and 3 units up.
Intercepts
Intercepts are where a line crosses the axes on a coordinate plane.

The x-intercept is where the line crosses the x-axis, meaning the y value is zero at this point. Conversely, the y-intercept is where the line crosses the y-axis, so the x value is zero here.

In the given equation y = 3x, both the x-intercept and y-intercept are at (0, 0). This means the line goes through the origin.
To find the intercepts:
  • Set x = 0 to find the y-intercept.
  • Set y = 0 to find the x-intercept.
Slope
The slope of a line explains its steepness and direction.

It's represented by 'm' in the linear equation y = mx + c. A positive slope means the line goes up from left to right, and a negative slope means it goes down.

The slope is typically calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In simpler terms, it's: m = (change in y) / (change in x).
In our equation y = 3x, the slope is 3. This means for every unit you move to the right on the x-axis, you'll move up 3 units on the y-axis.
Linear Equation
A linear equation forms a straight line when graphed on a coordinate plane.

It's usually written in the form y = mx + c, where:
  • y is the dependent variable (output)
  • x is the independent variable (input)
  • m is the slope
  • c is the y-intercept
Linear equations can have one variable or two, and they show a constant rate of change. In the equation y = 3x, the y-intercept is 0 and the slope is 3, simplifying it to y = 3x, indicating a line passing through the origin with a steepness of 3 units up for every unit right.

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Most popular questions from this chapter

Round off to the nearest hundredth when necessary. A gourmet food shop sells custom blended coffee by the ounce. Suppose that 1 oz sells for 1.70 dollar and 2 oz sells for 3.20 dollar Assume that the cost \(c\) of the coffee is linearly related to the number of ounces \(n\) purchased (where \(n \geq 1\) ). (a) Write an equation relating the cost of the coffee to the number of ounces purchased. (b) What would be the cost of 3.5 oz of coffee? (c) Suppose a package of coffee is marked at 6.50 dollar but has no indication of how much coffee it contains. Determine the number of ounces of coffee this package contains. (d) Sketch a graph of this equation for \(n \geq 1\) using the horizontal axis for \(n\) and the vertical axis for \(c .\)

This exercise discusses the relationship between the slopes of perpendicular lines. (a) Sketch the graphs of \(y=2 x+4\) and \(y=-\frac{1}{2} x+4\) on the same coordinate system. (b) On the basis of your graph, does it appear that these lines are perpendicular? (c) What is the relationship between the slopes of these two lines? (d) It is a fact that the slopes of perpendicular lines are negative reciprocals of each other (provided that neither of the lines is vertical). What is the slope of a line perpendicular to the line whose equation is \(y=\frac{2}{5} x+7 ?\)

Determine the slope of the line from its equation. $$2 x-5 y+7=0$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. On a certain history exam, \(46 \%\) of the questions were multiple-choice questions, 5 were essay questions, and the remaining \(44 \%\) were fill-in-the- blank questions. How many multiple-choice questions were there?

Determine the slope of the line from its equation. $$3 y-5 x=12$$
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