Problem 81 - College Algebra

Question

This 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

Answer

The graph of the linear function crosses the x-axis at the point (3, 0) and the y-axis at the point (0, 2).

Step 1: Finding the x-intercept

Set \( y = 0 \), and solve for \( x \) in the equation. This will return \( x = 3 \).

Step 2: Finding the y-intercept

Now, set \( x = 0 \), and solve for \( y \) in the equation. This will return \( y = 2 \).

Step 3: Drawing the Graph

Plot the x and y intercepts on the graph. The x-intercept is at (3, 0) and the y-intercept is at (0, 2). Draw a straight line between these two points. The line represents the graph of the given linear function.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a foundational skill in mathematics, helping visualize the relationship between variables. A linear equation represents a straight line on a graph. To graph a linear equation, you typically need at least two points. Commonly used points are the x-intercept and y-intercept because they are easy to calculate. Once you have these points, plotting them on a Cartesian plane becomes straightforward. Connect the dots, and you have your linear function visibly represented. Thanks to this method, analyzing and understanding how changes in variables affect each other becomes intuitive.

X-intercept

The x-intercept is the point where the graph of a linear equation crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept from an equation, substitute 0 for y and solve for x. For example, in the equation \(2x - 3y = 6\), setting \(y = 0\) gives \(2x = 6\). Solving for x results in \(x = 3\). This means the x-intercept is at the point (3, 0). Knowing the x-intercept helps in understanding the behavior of the line as it approaches the x-axis.

Y-intercept

The y-intercept is where a line crosses the y-axis. At the y-intercept, the value of x is zero. To find it, set x to zero in the equation and solve for y. In the example \(2x - 3y = 6\), substituting \(x = 0\) gives \(-3y = 6\). Solving this yields \(y = -2\), meaning the y-intercept is (0, -2). The y-intercept allows you to determine where the line starts or intersects the y-axis, which is crucial for accurately plotting the line on a graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is handy because it provides immediate insight into the line's direction and position. The slope \(m\) indicates how steep the line is and the direction of the incline. The y-intercept \(b\) shows where the line crosses the y-axis. Converting a standard linear equation to slope-intercept form makes graphing simpler. For example, by rearranging the equation \(2x - 3y = 6\) into \(y = \frac{2}{3}x - 2\), you can easily identify the slope and y-intercept to aid in graphing.