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Chapter 4: Problem 317

Graph the line of each equation using its slope and \(y\) -intercept. \(4 x-3 y=6\)

Short Answer

Expert verified
First, convert the equation to slope-intercept form as \(y = \frac{4}{3}x - 2\).Then, plot the y-intercept (-2) and use the slope \(\frac{4}{3}\) to find another point. Finally, draw the line through these points.

Step by step solution

01

- Write the equation in slope-intercept form

To graph the line, first write the equation in the slope-intercept form which is \(y = mx + b\). Solve for \(y\) in the given equation. Start with the equation:4x - 3y = 6.Subtract 4x from both sides to isolate the term with \(y\):-3y = -4x + 6.Next, divide every term by -3 to solve for \(y\):\(y = \frac{4}{3}x - 2\).
02

- Identify the slope and y-intercept

The slope-intercept form of the line is now \(y = \frac{4}{3}x - 2\).From this equation, identify the slope (\(m\)) and the y-intercept (\(b\)).Here, the slope \(m = \frac{4}{3}\) and the y-intercept \(b = -2\).
03

- Plot the y-intercept

On a graph, start by plotting the y-intercept. The y-intercept is where the line crosses the y-axis. For our equation, the y-intercept is -2. So plot the point (0, -2) on the y-axis.
04

- Use the slope to find another point

The slope of \(\frac{4}{3}\) means that for every 3 units you move to the right on the x-axis, you move up 4 units on the y-axis. From the y-intercept (0, -2), move 3 units to the right (to x = 3) and then move up 4 units to y = 2. Plot the point (3, 2).
05

- Draw the line

Once you have two points, draw a straight line through these points to represent the equation \(4x - 3y = 6\).Extend the line in both directions and add arrows on both ends to show it continues infinitely.

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Key Concepts

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

Slope-intercept form
Understanding the slope-intercept form is key to graphing linear equations with ease. The slope-intercept form of a linear equation is written as:
\[ y = mx + b \]
In this form:
  • \((m)\) represents the slope of the line.
  • \((b)\) represents the y-intercept of the line.
By converting a linear equation into this form, you can quickly identify both the slope and the y-intercept. This makes the process of graphing much more straightforward. For instance, in the given equation, \((4x - 3y = 6)\), we rearranged it to get \((y = \frac{4}{3}x - 2)\).
Plotting points
Plotting points on a graph helps in visualizing the line represented by the equation. Once you have the slope-intercept form of an equation, start by plotting the y-intercept. This is the point where the line crosses the y-axis, making it your starting point for graphing. After plotting the y-intercept, use the slope to find additional points on the graph. By moving horizontally (along the x-axis) and vertically (along the y-axis) as indicated by the slope, you can find another point on the line. Finally, draw a straight line through these points to represent the equation.

Y-intercept
The y-intercept is the point where the line crosses the y-axis. It can be quickly identified from the slope-intercept form \((y = mx + b)\) as the constant term, \((b)\). In our example, the y-intercept is \((b = -2)\). To graph this, simply locate -2 on the y-axis and plot the point \((0, -2)\). This point serves as the starting reference for graphing the line.
Slope
The slope of a line measures its steepness and direction. It is determined by the value of \((m)\) in the slope-intercept form \((y = mx + b)\). In this context, slope is the ratio of the rise (vertical change) to the run (horizontal change). For instance, in our equation \((y = \frac{4}{3}x - 2)\), the slope is \((\frac{4}{3})\). This means that for every 3 units we move to the right on the x-axis, we move up 4 units on the y-axis. From the y-intercept \((0, -2)\), move 3 units to the right to \((x = 3)\) and then move up 4 units to \((y = 2)\). Plot this point \((3, 2)\) to help graph the line accurately.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-3,-4) and (2,-5)

Find the equation of each line. Write the equation in slope-intercept form. \(m=\frac{5}{6},\) containing point (6,7)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-1,3) and (-6,-7)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=4 x+2,\) point (1,2)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(x-2=0,\) point(1,-2)
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