Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{aligned} 2.4 x+3.8 y &=-17.6 \\ 4 x-0.2 y &=-3.2 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The solution of the system is approximately (x, y), where x and y coordinates are obtained and rounded to three decimal places from the intersection point.

Step 1: Graphing the first equation

Firstly, express the first equation , \(2.4x+3.8y = -17.6\), into the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Thus, the equation will become: \(y = (-2.4/3.8)x - (17.6/3.8)\). Then graph this equation using a graphing utility.

Step 2: Graphing the second equation

Next, convert the second equation, \(4x - 0.2y = -3.2\), also into the slope-intercept form. It would become: \(y = (4/0.2)x + (-3.2/0.2)\). After that, graph this equation on the same grid as the first one.

Step 3: Finding the intersection point

After both equations are graphed, look for the point where the two lines intersect. This intersection point represents the solution of this system of equations. Use the graphing utility to approximate the coordinates of the intersection point.

Step 4: Rounding the result

Round the obtained x and y values of the intersection point to three decimal places to get the final solution.

Key Concepts

Graphing Utility

When working with equations, a graphing utility is an incredibly helpful tool for visualizing relationships between variables. It's a digital tool, like a software or a feature on a graphing calculator, that can take an equation and produce its graph on a coordinate plane with minimal effort from the user.

Graphing utilities typically require the input of an equation in a particular form. Once the equations are entered, the utility quickly plots them, showing their shape, position, and intersection points. For students, using a graphing utility can make it easier to understand where solutions lie within a system of equations without having to plot points manually.

For the given exercise, employing a graphing utility helps approximate the solution by providing a visual representation of where two lines would intersect, representing the solution to the system of equations.

Slope-Intercept Form

In algebra, one of the most convenient ways to express a linear equation is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) symbolizes the slope of the line, and \(b\) represents the y-intercept, where the line crosses the y-axis.

Converting an equation to this form involves solving for \(y\) and simplifying until you have isolated \(y\) on one side. This is especially useful for graphing, as it allows you to identify the initial point of the line on the y-axis and the direction and steepness determined by the slope. In the given exercise, both equations are converted into this form to prepare them for graphing, thus simplifying the process of finding the graphical solution.

Graphical Solution of System

A graphical solution to a system of equations involves finding where two or more lines intersect on a graph. Each equation in the system represents a line on the coordinate plane. The point of intersection is the set of coordinates that satisfy all equations simultaneously—it is the solution to the system.

For visual learners, this method can be particularly intuitive as it transforms abstract equations into a concrete visual where the answer is the point where the lines cross. It's worth noting that if lines are parallel (indicating no solutions) or identical (indicating infinite solutions), this will also be apparent from the graph. In the context of our exercise, after graphing each equation using the slope-intercept form, the graphing utility provides a clear visual representation to identify the intersection point.

Intersection Point of Lines

The intersection point of lines is the heart of solving a system of equations graphically. It is the precise spot where the lines that represent the equations cross. The coordinates of this point satisfy all the equations in the system.

Finding this point visually can be easier and more direct, although less precise than algebraic methods. A graphing utility often has functionalities to approximate the intersection point, guiding you towards the answer. In our exercise example, the graphing utility will enable us to estimate the point of intersection, and from there, we can round the coordinates as required to complete the solution process.