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Chapter 11: Problem 242

Use the intercept form to graph \(3 \mathrm{x}+4 \mathrm{yx}=12\).

Short Answer

Expert verified
The x-intercept of the given equation \(3x + 4yx = 12\) is at the point (4, 0), and the y-intercept is at the point \(\left(0, \frac{12}{4y}\right)\), considering that y is not 0. To graph the equation, plot the x-intercept, the various points for the y-intercept as y 鈮�0, and draw a straight line that passes through them.

Step by step solution

01

Write the equation

Given the equation: \(3x + 4yx = 12\).
02

Find x-intercept (y = 0)

To find the x-intercept, we set y = 0 in the equation and solve for x. \(3x + 4(0)x = 12\) This simplifies to: \(3x = 12\) Now, divide both sides by 3: \(x = 4\) So, the x-intercept is at the point (4, 0).
03

Find y-intercept (x = 0)

To find the y-intercept, we set x = 0 in the equation and solve for y. \(3(0) + 4yx = 12\) This simplifies to: \(4yx = 12\) Now, divide both sides by 4y (assuming y is not 0): \(x = \frac{12}{4y}\) So, the y-intercept is at the point \(\left(0, \frac{12}{4y}\right)\), considering that y is not 0 (since we cannot divide by 0).
04

Graph the equation

Now, we can plot the x and y intercepts on the graph. 1. Plot the x-intercept found in step 2: (4, 0). 2. Plot the various points for the y-intercept found in step 3: \(\left(0, \frac{12}{4y}\right)\), as y 鈮�0. 3. Draw a straight line that passes through the x-intercept and the various y-intercept points plotted. The graph of the equation \(3x + 4yx = 12\) is now complete.

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Key Concepts

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

X-Intercept
The x-intercept of a graph represents the point where the line crosses the x-axis. In the context of our equation, to find it, we let y equal zero and solve for x. This intercept can be thought of as the point where the output of the function (y) is zero. In our example, the x-intercept was found to be 4, meaning when y is zero, x is 4. Visually on a graph, this is the point (4, 0), situated along the x-axis. Understanding the x-intercept helps us to start plotting our graph and provides an essential anchor point for drawing the line of the equation.
Y-Intercept
Conversely, the y-intercept is where the line crosses the y-axis. This is found by setting x to zero and solving for y. The y-intercept indicates the value of y when the input x is zero. For many linear equations, this is a single point, but in the given equation, the presence of yx suggests multiple y-intercept points, a topic that may require further exploration beyond a typical linear equation. Generally, for a simple linear equation, once you've found the y value, you plot the point on the y-axis, usually written as (0, y).
Intercept Form
The intercept form is a convenient way to write linear equations that highlights the x and y intercepts, usually of the form \( x/a + y/b = 1 \). This form makes it particularly easy to spot and plot the x and y intercepts on the graph, which are (a, 0) and (0, b), respectively. When dealing with the provided equation, the intercept form could be a means to solving the equation. However, intercept form would not apply straightforwardly due to the yx term. This illustrates a key strategy: adapting forms and methods based on the characteristics of specific equations.
Plotting Intercepts
Plotting intercepts is the foundational step in graphing linear equations. It involves placing a dot at the x-intercept and the y-intercept on a coordinate plane and then drawing a line through these two points to represent the equation graphically. While most linear equations will result in a single x-intercept and a single y-intercept, certain equations, like the one in our example, may hint at more complexity. Nonetheless, by focusing on intercepts, even beginning graphers can swiftly move towards an accurate representation of a line on a graph.

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Most popular questions from this chapter

Graph the function defined by \(3 \mathrm{x}-4 \mathrm{y}=12\)

Construct the graph of the function defined by \(\mathrm{y}=3 \mathrm{x}-9\).

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Discuss the graph of the function \(\mathrm{y}=-3 \mathrm{x}+4\).

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