Problem 76 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}2 x-3 y=10 \\ 4 x+3 y=20\end{array}\right.$$

Step-by-Step Solution

Answer

By graphing the lines, you would ideally find one point of intersection which represents the solution to the system of equations. However, note that the result may vary slightly as it may be affected by the precision of the graphing utility used.

Step 1: Convert equations into slope-intercept form

For the first equation, \(2x - 3y = 10\), divide all terms by -3 to isolate y: \(y = -\frac{2}{3}x + \frac{10}{3}\). For the second equation, \(4x + 3y = 20\), divide all terms by 3 to isolate y: \(y = -\frac{4}{3}x + \frac{20}{3}\).

Step 2: Plot lines using a graphing utility

Using a graphing utility, plot the two equations \(y = -\frac{2}{3}x + \frac{10}{3}\) and \(y = -\frac{4}{3}x + \frac{20}{3}\). These are two straight lines, and their intersection would give the solution to the system.

Step 3: Find intersection of the lines

The solution is at the point where the two lines intersect. Use the graphing utility to find this intersection point.

Key Concepts

Slope-Intercept Form

The slope-intercept form of an equation provides a straightforward way to graph a linear equation. It's written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) the y-intercept. For the equations in this exercise, the task is to rearrange them into this form.

For the equation \(2x - 3y = 10\), we solve for \(y\) by isolating it on one side. This gives us the equation \(y = -\frac{2}{3}x + \frac{10}{3}\). Here, \(-\frac{2}{3}\) is the slope, indicating the line descends by 2 units for every 3 units it moves horizontally. The \(\frac{10}{3}\) is the y-intercept, where the line crosses the y-axis.

Similarly, transforming \(4x + 3y = 20\) into \(y = -\frac{4}{3}x + \frac{20}{3}\), helps in graphing. The slope \(-\frac{4}{3}\) suggests a steeper decline than the first equation, and the \(\frac{20}{3}\) is where it touches the y-axis. Understanding these transformations is crucial for plotting equations effectively.

Graphing Utility

A graphing utility is a helpful tool, especially when dealing with systems of equations. It swiftly visualizes complex equations, allowing users to comprehend the relationships between lines. In this exercise, it's used to plot two lines, aiding in the identification of their intersection points.

When you input the equations \(y = -\frac{2}{3}x + \frac{10}{3}\) and \(y = -\frac{4}{3}x + \frac{20}{3}\) into a graphing utility, it displays them as diagrams on the coordinate plane. This visual representation provides clear insights into how each line behaves and their potential points of interaction.

Using such a utility not only saves time but also minimizes calculation errors, as it renders exact plots of complex systems. This is particularly beneficial when solving systems with more than two variables. It allows you to easily adjust, analyze, and interpret various components of the equations.

Intersection of Lines

In systems of equations, finding the intersection point is key, as it represents the solution. It's the point where both equations satisfy at the same time. Graphically, this is where the lines from each equation meet or intersect.

After plotting the equations \(y = -\frac{2}{3}x + \frac{10}{3}\) and \(y = -\frac{4}{3}x + \frac{20}{3}\), the intersection gives the exact values of \(x\) and \(y\) that solve the system. Plugging these values back into the original equations verifies that both conditions hold true.

If the lines intersect at a single point, the system of equations has a unique solution. If the lines were parallel, they would never meet, indicating no solution. Alternatively, if the lines lie on top of each other, the system would have infinitely many solutions. Thus, assessing the nature of the intersection helps in understanding the dynamics of the system thoroughly.