Problem 81
Question
will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ 2 x-3 y=6 $$
Step-by-Step Solution
Verified Answer
The graph of the function \(2x - 3y = 6\) is a straight line with a slope of \(\frac{2}{3}\) and y-intercept of -2.
1Step 1: Convert to Slope-Intercept Form
To convert the equation \(2x - 3y = 6\) to slope-intercept form, isolate y. This can be done by first subtracting 2x from both sides of the equation to obtain \(-3y = -2x + 6\), then dividing everything by -3 to get \(y = \frac{2}{3}x - 2\)
2Step 2: Identify the Slope and the Y-Intercept
Now that the equation is in slope-intercept form, we can identify the slope (m) and the y-intercept (b). From the equation, we can see that \(m = \frac{2}{3}\) and \(b = -2\). The slope \(\frac{2}{3}\) indicates that for every 3 units moved horizontally (either left or right), the line will move 2 units vertically (up if moving to the right, down if moving to the left). The y-intercept -2 indicates that the line crosses the y-axis at the point (0, -2).
3Step 3: Sketch the Line
First, plot the y-intercept on the graph at the point (0, -2). Then, using the slope, move 3 units to the right and 2 units up from the y-intercept and make another point. Drawing a straight line through these points will give the graph of the linear function.
Key Concepts
Graphing Linear EquationsSlope-Intercept FormSlope and Y-Intercept
Graphing Linear Equations
Graphing linear equations is a valuable skill that allows you to visually represent mathematical relationships. To graph a linear equation, you first need to understand what the equation represents and how to turn it into a graph.
One of the first steps in graphing is recognizing the type of equation you are working with, typically given in a format known as the standard form. In this exercise, we start with \(2x - 3y = 6\). To easily graph this, it's best to convert it into slope-intercept form, where you'll have the equation in the form \(y = mx + b\). This makes it much easier to see how the line will appear on a graph.
Now that we have our equation in the form \(y = \frac{2}{3}x - 2\), we can get to graphing. Begin by plotting the y-intercept on the graph—and here it's \(-2\) on the y-axis. From this point, use the slope to determine the other points on the line. A handy trick for accuracy is to continue using the slope to plot several points before drawing your line through them. This ensures a straight, precise result.
One of the first steps in graphing is recognizing the type of equation you are working with, typically given in a format known as the standard form. In this exercise, we start with \(2x - 3y = 6\). To easily graph this, it's best to convert it into slope-intercept form, where you'll have the equation in the form \(y = mx + b\). This makes it much easier to see how the line will appear on a graph.
Now that we have our equation in the form \(y = \frac{2}{3}x - 2\), we can get to graphing. Begin by plotting the y-intercept on the graph—and here it's \(-2\) on the y-axis. From this point, use the slope to determine the other points on the line. A handy trick for accuracy is to continue using the slope to plot several points before drawing your line through them. This ensures a straight, precise result.
Slope-Intercept Form
The slope-intercept form is a popular way to express linear equations. It is written as \(y = mx + b\) and clearly shows both the slope of the line and its y-intercept.
The slope \(m\) represents how steep the line is. It tells you how much \(y\) will change with every step you take along the \(x\)-axis. For example, in the equation \(y = \frac{2}{3}x - 2\), the slope \(\frac{2}{3}\) means when you move 3 units along the x-axis, you should move 2 units up or down on the y-axis, depending on the direction you're moving.
The y-intercept \(b\) is the point where the line crosses the y-axis. This makes the slope-intercept form particularly useful because with just one glance, you can see where to start your graph—by plotting the y-intercept—and how to continue it using the slope.
The slope \(m\) represents how steep the line is. It tells you how much \(y\) will change with every step you take along the \(x\)-axis. For example, in the equation \(y = \frac{2}{3}x - 2\), the slope \(\frac{2}{3}\) means when you move 3 units along the x-axis, you should move 2 units up or down on the y-axis, depending on the direction you're moving.
The y-intercept \(b\) is the point where the line crosses the y-axis. This makes the slope-intercept form particularly useful because with just one glance, you can see where to start your graph—by plotting the y-intercept—and how to continue it using the slope.
- The y-intercept makes it easy to start your line graph immediately.
- The slope provides a consistent pattern for drawing the line.
Slope and Y-Intercept
Understanding the slope and y-intercept plays a key role in effectively graphing a linear equation.
The **slope** is a critical component, symbolized as \(m\) in \(y = mx + b\). It defines the angle and direction of the line. A positive slope means the line inclines as it moves from left to right, while a negative slope would decline.
In our example, \(m = \frac{2}{3}\). This indicates that the line goes up 2 units for every 3 units you move to the right, creating an upwards slope.
The **y-intercept** \(b\) is equally important. It tells you where to begin plotting the line on a graph—the point where the line crosses the y-axis. Here, \(b = -2\), so you start drawing your line at the coordinate (0, -2).
The **slope** is a critical component, symbolized as \(m\) in \(y = mx + b\). It defines the angle and direction of the line. A positive slope means the line inclines as it moves from left to right, while a negative slope would decline.
In our example, \(m = \frac{2}{3}\). This indicates that the line goes up 2 units for every 3 units you move to the right, creating an upwards slope.
The **y-intercept** \(b\) is equally important. It tells you where to begin plotting the line on a graph—the point where the line crosses the y-axis. Here, \(b = -2\), so you start drawing your line at the coordinate (0, -2).
- The y-intercept is your starting point on the graph at \(x = 0\).
- The slope will guide your direction and steepness.
Other exercises in this chapter
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