Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Graph each line by hand. Give the \(x\) - and y-intercepts. $$2 x-3 y=6$$

Step-by-Step Solution

Answer

The x-intercept is (3, 0) and the y-intercept is (0, -2).

Step 1: Identify the Equation of the Line

We are given the equation in the standard form: \(2x - 3y = 6\). This equation is in the form \(Ax + By = C\), where \(A = 2\), \(B = -3\), and \(C = 6\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(2x - 3y = 6\). This gives \(-3y = 6\), solving for \(y\) gives \(y = -2\). The y-intercept is \((0, -2)\).

Step 3: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation \(2x - 3y = 6\). This results in \(2x = 6\), solving for \(x\) gives \(x = 3\). The x-intercept is \((3, 0)\).

Step 4: Plot the Intercepts on a Graph

On a graph with the x-axis and y-axis, plot the points \((0, -2)\) and \((3, 0)\). These points are the y-intercept and x-intercept, respectively.

Step 5: Draw the Line

Using a ruler, draw a straight line through the points \((0, -2)\) and \((3, 0)\). This line represents the graph of the equation \(2x - 3y = 6\).

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of y is always zero.
To find the x-intercept of a linear equation, you substitute 0 in place of y and solve for x.
In the equation provided, which is in the standard form, you have: \[2x - 3y = 6\]Here's how to find the x-intercept step-by-step:

• Set y to 0, because that's the value at the x-intercept: \[2x = 6\]
• Solve for x: \[x = \frac{6}{2} = 3\]
• So, the x-intercept is the point (3, 0).

Remember, recognizing the x-intercept is crucial because it provides one of the key points needed to draw the graph accurately.

y-intercept

The y-intercept is simply where the line crosses the y-axis, which happens when x equals zero.
This tells you at what point the line reaches the y-axis, and no matter what the line is or its direction, the y-coordinate of this point tells you how high or low the line goes on this axis.
In our linear equation, you will find the y-intercept by following these steps:

• Set x to 0 in the equation: \[2(0) - 3y = 6\]
• Solve the equation for y: \[-3y = 6\]
• Divide both sides by -3 to isolate y: \[y = -2\]
• Hence, the y-intercept is the point (0, -2).

The y-intercept is important because it's an easy way to start graphing a line since it's where the line begins on the y-axis.

standard form of a line

The standard form of a linear equation is one way of writing the equation of a line. This form is characterized by its structure: \[Ax + By = C\]
where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) cannot both be zero.

• The coefficients \(A\) and \(B\) tell you about the slope, and the constant \(C\) can help determine the intercepts.
• For graphing, it's often helpful to convert or utilize this form to find intercepts easily.

In our exercise, the equation \[2x - 3y = 6\] is already in standard form:

• Here, \(A = 2\), \(B = -3\), and \(C = 6\).

Understanding the standard form is crucial because it allows you to see how x and y relate in terms of their coefficients. This form is especially useful when you need to quickly find the intercepts to start graphing.