Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$4 x-2 y=-40$$

Step-by-Step Solution

Answer

The equation \(y = 2x + 20\) is graphed. The x-intercept is identified at (-10,0) and the y-intercept is identified at (0,20).

Step 1: Rearrange the equation in the form of \(y = mx + c\)

For the equation \(4x - 2y = -40\), the first step would be to bring it into the standard form (y = mx + c). This can be achieved by isolating the variable \(y\). You can divide the entire equation by 2 to simplify: \(2x - y = -20\). Now, add \(y\) to both sides of the equation and subtract 20 from both sides to get: \(y = 2x + 20\).

Step 2: Graph the equation

The equation can now be graphed using a graphing utility. On the graph, plot the equation \(y = 2x + 20\). This would produce a straight line.

Step 3: Identify the x-intercept and y-intercept

The x-intercept is the point where the line crosses the x-axis (i.e., when \(y = 0\)). Putting \(y = 0\) in the equation, you get \(x = -10\), so the x-intercept is at (-10,0). The y-intercept is where the line crosses the y-axis (when \(x = 0\)). Putting \(x = 0\) in the equation, you get \(y = 20\), so the y-intercept is at (0,20).

Key Concepts

Equation Rearrangement

Rearranging an equation is one of the fundamental skills in algebra. We aim to solve linear equations in terms of a single variable. This involves expressing the equation in the slope-intercept form as much as possible. For our exercise, we started with the equation: \(4x - 2y = -40\). The goal is to isolate the variable \(y\).
To do this, we first simplified the coefficients by dividing the whole equation by 2. This step gave us: \(2x - y = -20\).
Next, we added \(y\) to both sides of the equation to avoid negatives, delivering \(2x = y - 20\).
The final move was to add 20 to both sides, isolating \(y\) on one side, thus:

• \(y = 2x + 20 \)

This form, \(y = mx + c\), is crucial as it provides a clear view of the linear equation's dynamics for easy graphing.

Identifying Intercepts

Intercepts are the points where the line crosses the axes. Understanding these points is vital to comprehending the behavior of the equation graphically. Let's break down finding the intercepts:

• **X-Intercept**: This is where the line cuts the x-axis. For this point, \(y = 0\). Using \(y = 2x + 20\) and setting \(y = 0\), we find \(x\) by solving \(0 = 2x + 20\), which results in \(x = -10\). Thus, the x-intercept is \((-10, 0)\).
• **Y-Intercept**: This is where the line intersects the y-axis. Here, \(x = 0\). Substituting \(x = 0\) into the equation \(y = 2(0) + 20\), we find \(y = 20\). Therefore, the y-intercept is \(0, 20\).

Intercepts are vital because they give two direct reference points to start graphing a linear equation.

Graphing Utilities

Graphing utilities are technological tools that help in visualizing mathematical expressions. These can include graphing calculators and various online tools. They make graph-related tasks quicker and more intuitive, especially when dealing with complicated equations.
Here’s a typical process to follow:

• First, ensure the equation is solved for \(y\) in terms of \(x\) (as in \(y = 2x + 20\)).
• Input the equation into the graphing utility.
• Choose the window settings to view an appropriate scale.

The graphing utility will generate a line on a coordinate grid representing the equation \(y = 2x + 20\).
One powerful feature of graphing utilities is their ability to find intercepts, simply by tracing the curve and observing where it crosses the axes.

Algebraic Manipulation

Algebraic manipulation involves using arithmetic and algebraic operations to solve equations or simplify expressions. In our scenario, algebraic manipulation allows us to rearrange the equation and prepare it for graphing.
Here are a few techniques we used:

• **Division to Simplify**: Dividing the entire equation \(4x - 2y = -40\) by 2 helped in reducing large coefficients.
• **Adding/Subtracting Terms**: After division, we had \(2x - y = -20\). To solve for \(y\), we needed to reorganize terms by adding \(y\) to both sides and later \(20\) to isolate \(y\).
• **Solving for Variables**: Algebraic manipulation integrated with identifying intercepts allowed us to plug in \(x = 0\) and \(y = 0\) to find intercepts. This underscores the versatility of algebra in applying wide-ranging math tasks.

Mastering algebraic manipulation not only helps line up an equation for graphing but also enhances problem-solving proficiency, vital for handling diverse mathematical challenges.