Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Graph each line by hand. Give the \(x\) - and y-intercepts. $$4 x-3 y=9$$

Step-by-Step Solution

Answer

The x-intercept is \(\left(\frac{9}{4}, 0\right)\) and the y-intercept is \((0, -3)\).

Step 1: Identify the Problem

We are given the linear equation \(4x - 3y = 9\). To graph this line, we need to find both the \(x\)-intercept and the \(y\)-intercept.

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\):\[4x - 3(0) = 9\]\[4x = 9\]\[x = \frac{9}{4}\]So, the \(x\)-intercept is at \(\left(\frac{9}{4}, 0\right)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\):\[4(0) - 3y = 9\]\[-3y = 9\]\[y = -3\]Thus, the \(y\)-intercept is at \((0, -3)\).

Step 4: Plot the Intercepts

Use the intercepts \(\left(\frac{9}{4}, 0\right)\) and \((0, -3)\) to plot points on the graph. These two points determine the line.

Step 5: Draw the Line

Draw a straight line through the two intercept points to complete the graph of the equation.

Key Concepts

Understanding x-intercepts

The x-intercept is a critical concept when graphing linear equations. It represents the point where a graph crosses the x-axis. At this point, the value of y is always zero.

• To find the x-intercept, substitute 0 for y in the equation and solve for x. For example, in the equation \(4x - 3y = 9\), if we set \(y = 0\), we get:\[4x = 9\]
• Simplifying yields \(x = \frac{9}{4}\), giving the intercept point as \(\left(\frac{9}{4}, 0\right)\).
• This intercept is often plotted on the graph as it provides an anchor point.

Recognizing and calculating this point is essential for accurately graphing the line.

Understanding y-intercepts

Similar to the x-intercept, the y-intercept is where the graph crosses the y-axis, meaning the x value is zero at this point.

• To find the y-intercept, substitute 0 for x in the equation and solve for y. For example, in the equation \(4x - 3y = 9\), substituting \(x = 0\) results in \(-3y = 9\).
• Solving this equation gives \(y = -3\), indicating the intercept at \((0, -3)\).
• This y-intercept is key for graphing because it serves as another crucial reference point.

Utilizing both intercepts allows for the drawing of a straight line, representing the linear equation on the graph.

The character of linear equations

A linear equation is any equation that can be written in the form of \(ax + by = c\), where a, b, and c are constants.

• These equations consistently graph as a straight line when plotted on a coordinate plane.
• The general characteristics of linear equations include a constant rate of change and a graph that doesn't curve or break.
• Every linear equation has at least one solution and can have two intercepts, specifically the x- and y-intercepts.

Understanding these characteristics helps in recognizing the solutions' extent and how to approach plotting them.

Mastering graphing techniques for linear equations

Graphing linear equations efficiently is a fundamental skill in mathematics. It involves utilizing a few systematic techniques:

• First, calculate both the x- and y-intercepts – these are the primary steps as they provide key points through which the line will pass.
• Plotting these intercepts on the coordinate plane is crucial because they ease the visualization process.
• Once plotted, simply draw a straight line through these points. Ensure the line extends in both directions to demonstrate that it continues infinitely.
• Also, understanding the slope can help. It represents the steepness of the line, calculated as the change in y over the change in x.

By mastering these techniques, you'll efficiently map out linear equations, making them easier and faster to analyze.