Problem 3 - Technical Mathematics with Calculus

Question

Graphically find the approximate solution to each system of equations. If you have a graphics calculator, use the \([\mathrm{ZOOM}]\) and \(\mathrm{TRACE}\). or \([\text { INTERSECT }]\) features to find the solution. $$\begin{aligned} &x-2 y=-3\\\ &3 x+y=5 \end{aligned}$$

Step-by-Step Solution

Answer

The approximate solution to the system of equations is the intersection point of the two lines when graphed. Use a graphing calculator's features for an exact solution.

Step 1: Plot the first equation

Convert the first equation, \(x - 2y = -3\), to slope-intercept form to determine the y-intercept and the slope. This form is \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. The equation becomes \(y = \frac{1}{2}x + \frac{3}{2}\). Plot this line on a graph using the y-intercept and slope, or enter it into the graphing calculator.

Step 2: Plot the second equation

Convert the second equation, \(3x + y = 5\), to slope-intercept form to find the slope and y-intercept. It becomes \(y = -3x + 5\). Plot this line using the y-intercept and slope, or enter it into the graphing calculator.

Step 3: Identify Points of Intersection

Look for the point or points where the two lines intersect. This point of intersection represents the solution to the system of equations.

Step 4: Use Graphing Calculator Features

If using a graphing calculator, employ the \(\mathrm{ZOOM}\) and \(\mathrm{TRACE}\) features to better visualize the point of intersection or use the \(\mathrm{INTERSECT}\) feature to find the exact coordinates of the intersection point.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphically solving systems of equations. In this form, any linear equation can be written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. The slope quantifies the steepness and direction of the line; a positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. For the first line in our exercise, converting \( x - 2y = -3 \) to slope-intercept form reveals a slope of \( \frac{1}{2} \), indicating it rises at a half unit up for every one unit right, with a y-intercept of \( \frac{3}{2} \). Similarly, the second equation, when converted to \( y = -3x + 5 \), shows a negative slope of -3, suggesting a steep decline, and intersects the y-axis at 5. Graphing these equations using their slope and y-intercept makes discerning their relationship - and the system's solution - much easier.

Graphing Calculator Features

A graphing calculator is a powerful tool for visualizing and solving systems of equations. Its features can greatly simplify the process of finding the solution graphically. The \( \mathrm{ZOOM} \) function allows you to alter the view of your graph, getting a closer look at particular sections where the lines might intersect, which can be very helpful if the intersection is not clearly visible with the default settings. The \( \mathrm{TRACE} \) function can be used to move along the curve of the graphs and find approximate coordinates of any point, including the points of intersection. By tracing, you can estimate the point where the equations intersect even if it doesn't snap exactly to the grid lines on the calculator's screen. Possibly the most precise feature is the \( \mathrm{INTERSECT} \) option. It allows you to select two different graphs and then calculates the exact point or points where they meet. This feature takes the guesswork out of finding the solution to a system of equations, giving you the exact x and y values of the intersection point.

Points of Intersection

The points of intersection are essentially the backbone of solving systems of equations graphically. They represent the set of coordinates that satisfy all the equations involved simultaneously. In the context of the exercise, where we have two linear equations, their intersection point, if one exists, is the solution to the system. To find the intersection graphically, you plot both equations on the same set of axes and observe where they cross. From the step by step solution provided, after plotting both lines derived from the equations (after converting them to slope-intercept form), you see that they intersect at a particular point. This is the solution to the system. In some cases, there may be no solution if the lines are parallel, or infinitely many solutions if the lines lie on top of each other, indicative of the same line. However, if the lines do intersect, that single point embodies the solution—each coordinate pair satisfying both original equations.