/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Solve by graphing. $$ y=\fra... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 38

Solve by graphing. $$ y=\frac{2}{3} x-5 $$

Short Answer

Expert verified
Plot points (0, -5) and (3, -3); draw a line through them.

Step by step solution

01

- Identify the slope and y-intercept

The given equation is in slope-intercept form, which is written as y = mx + b, where m is the slope and b is the y-intercept. Here, the slope m = \( \frac{2}{3} \) and the y-intercept b = -5.
02

- Plot the y-intercept

Start by plotting the y-intercept on the graph. Since the y-intercept is -5, place a point at (0, -5) on the y-axis.
03

- Use the slope to find another point

From the point (0, -5), use the slope \( \frac{2}{3} \). This means for every 3 units you move to the right (positive x-direction), you move up by 2 units (positive y-direction). Plot the second point at (3, -3).
04

- Draw the line

Draw a straight line through the two points plotted: (0, -5) and (3, -3). This line represents the equation y = \( \frac{2}{3} \)x - 5.

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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 one of the most common ways to represent a linear equation. It is written as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope indicates how steep the line is, while the y-intercept is the point where the line crosses the y-axis.
In the equation \(y = \frac{2}{3} x - 5\), the slope \(m\) is \( \frac{2}{3} \), which means for every 3 units you move to the right along the x-axis, you move up by 2 units along the y-axis.
The y-intercept \(b\) is -5, so the line crosses the y-axis at the point (0, -5). Understanding these two components makes it easy to graph the linear equation.
plotting points
Plotting points is essential for graphing linear equations. It helps visualize the line on a coordinate plane.

Here鈥檚 how to plot points for our equation \( y = \frac{2}{3} x - 5 \):
  • First, plot the y-intercept: Locate -5 on the y-axis and place a point at (0, -5). This is where the line starts.
  • Next, use the slope to find another point: From (0, -5), apply the slope \( \frac{2}{3} \). Move 3 units right (positive x-direction) and 2 units up (positive y-direction) to reach (3, -3).
  • Place the second point at (3, -3) on the graph.
These two points are enough to draw a precise line representing the equation.
drawing lines
Drawing lines connects the plotted points, visually representing the equation.

After you have your points from the previous section:
  • Start by placing a ruler or a straightedge through the points (0, -5) and (3, -3).
  • Draw a straight line extending in both directions beyond these points.
  • This line is the graphical representation of the equation \( y = \frac{2}{3} x - 5 \).
This method helps you see how the equation behaves across all x-values, giving you a complete linear graph. Now, you can see the slope and the exact placement of the y-intercept on the coordinate plane.

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