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Chapter 5: Problem 7

Graph the equation. $$ 2 x-3 y=24 $$

Short Answer

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Question: Graph the equation \(2x - 3y = 24\), and identify the x-intercept, y-intercept, and slope. Answer: The graph of the equation \(2x - 3y = 24\) has an x-intercept at (12, 0), a y-intercept at (0, -8), and a slope of \(\frac{2}{3}\).

Step by step solution

01

Find the x-intercept and y-intercept

To find the x-intercept, we will set y = 0 and solve for x. To find the y-intercept, we will set x = 0 and solve for y. For the x-intercept, let y = 0: $$ 2x - 3(0) = 24 \\ 2x = 24 \\ x = 12 $$ The x-intercept is (12, 0). For the y-intercept, let x = 0: $$ 2(0) - 3y = 24 \\ -3y = 24 \\ y = -8 $$ The y-intercept is (0, -8).
02

Write the equation in slope-intercept form

The slope-intercept form of a linear equation is given by: $$ y = mx + b $$ Where m is the slope and b is the y-intercept. Let's isolate y on one side of the equation to write it in slope-intercept form: $$ 2x - 3y = 24 \\ -3y = -2x + 24 \\ y = \frac{2}{3}x - 8 $$ So, the slope (m) is \(\frac{2}{3}\) and the y-intercept (b) is -8, which confirms the results from Step 1.
03

Plot the x-intercept, y-intercept, and the line

Using the x-intercept (12, 0) and the y-intercept (0, -8), we can plot the line on a graph. First, plot the x-intercept (12, 0) and y-intercept (0, -8) as points on the graph. Next, draw a line connecting these points and extending it in both directions. This line represents the graph of the given equation: $$ 2x - 3y = 24 $$ By following these steps, the given equation is graphed successfully.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept is where a graph crosses the x-axis. This happens when the y-value is zero.
To find the x-intercept of a linear equation like \(2x - 3y = 24\), set \(y = 0\) and solve for \(x\).
Here's a quick breakdown of this process:
  • Plug \(y = 0\) into the equation: \(2x - 3(0) = 24\).
  • It simplifies to \(2x = 24\).
  • Solving for \(x\), we divide both sides by 2, giving \(x = 12\).
Thus, the x-intercept of this line is at the point (12, 0).
Remember, whenever you want to find the x-intercept, set \(y\) to zero, and solve for \(x\). This point tells you where the line touches the x-axis.
Finding the Y-Intercept
The y-intercept is where a graph crosses the y-axis. At this point, the value of \(x\) is zero.
To determine the y-intercept from the equation \(2x - 3y = 24\), set \(x = 0\) and solve for \(y\).
Here鈥檚 a clear path to find it:
  • Substitute \(x = 0\) into the equation: \(2(0) - 3y = 24\).
  • This simplifies to \(-3y = 24\).
  • Solving for \(y\), divide both sides by -3, resulting in \(y = -8\).
Therefore, the y-intercept is at point (0, -8).
The y-intercept is critical as it shows where the line will touch the y-axis and provides a starting point on the graph.
Exploring the Slope-Intercept Form
The slope-intercept form of a linear equation is a convenient way to express a line. It is written as \(y = mx + b\).
Here, \(m\) is the slope, measuring the steepness and direction of the line. \(b\) is the y-intercept, showing where the line meets the y-axis.
To convert the equation \(2x - 3y = 24\) into slope-intercept form, follow these steps:
  • Rearrange to isolate \(y\): \(-3y = -2x + 24\).
  • Divide everything by -3 to solve for \(y\): \(y = \frac{2}{3}x - 8\).
Now, you can see the slope is \(\frac{2}{3}\), which means the line rises 2 units for every 3 units it runs to the right. The y-intercept is \(-8\), confirming our earlier calculation.
This form is particularly useful for quickly graphing a line or understanding its properties at a glance.

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