Problem 59 - 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. \(\left\\{\begin{array}{r}0.5 x+2.2 y=9 \\ 6 x+0.4 y=-22\end{array}\right.\)

Step-by-Step Solution

Answer

The approximate solution is around (x, y), where x and y are the rounded coordinates of the intersection point to three decimal places. The exact values depend on the graphing utility's representation.

Step 1: Rewrite the equations in slope-intercept form

First, rearrange the equations into the form y = mx + b. For the first equation \(0.5x + 2.2y = 9\), subtract \(0.5x\) from both sides and divide by \(2.2\) to get \(y = -0.227x + 4.091\). Do the same for the second equation \(6x + 0.4y = -22\), subtract \(6x\) from both sides and divide by \(0.4\) to get \(y = -15x - 55\).

Step 2: Graph the equations

Use a graphing utility to graph the two equations. The graph will show two lines where one is steeper than the other.

Step 3: Determine the intersection point

Look for the point where the two lines intersect. This point is the solution to the system of equations because it satisfies both equations simultaneously.

Step 4: Approximate and round the solution

Using the graphing utility, the intersection point can be approximated. Afterwards, the rounded solution of the intersection point to three decimal places can be identified.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that you can easily identify the slope and the y-intercept. It is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
To convert an equation into this form, you must rearrange the terms to isolate \(y\) on one side of the equation. This involves moving all other terms to the opposite side of the equation, then solving for \(y\).

• Example: For the equation \(0.5x + 2.2y = 9\), subtract \(0.5x\) and divide by \(2.2\) to get \(y = -0.227x + 4.091\).


• Another Example: For \(6x + 0.4y = -22\), you would subtract \(6x\) and divide by \(0.4\) to get \(y = -15x - 55\).

Using the slope-intercept form makes it much easier to graph equations because you can quickly identify how steep the line is and where it crosses the y-axis. This is crucial for visualizing solutions in systems of equations.

Graphing Utility

A graphing utility is a digital tool that helps you plot mathematical functions and equations on a coordinate plane. These utilities can be in the form of software programs, apps, or even sophisticated calculators. They are particularly useful when dealing with complex equations that are difficult to visualize manually.
With a graphing utility, you input the equations you've transformed into slope-intercept form, and the tool automatically draws the corresponding lines.

• This tool is advantageous because it handles the intricacies of calculation and drawing, freeing you to focus on understanding the math.


• It allows for quick experimentation; you can change a variable and instantly see the effect.

For the system of equations we're working with, graphing utilities will readily display two lines, enabling you to immediately see where they intersect, thus simplifying the task of finding a solution to the system.

Intersection Point

The intersection point is where two or more graphs meet on the coordinate plane. In the context of systems of equations, the intersection represents a common solution that satisfies all equations in the system.
Identifying the intersection point of two graphed lines involves looking for the exact coordinates where the lines cross.

• In a perfect mathematical world, this would yield an exact solution, but in real-world applications, graphing may require estimating.


• With a graphing utility, you can pinpoint this intersection more precisely and even approximate it to a certain number of decimal places.

In our example, the two equations you put into the graphing utility intersect at a point that, once found, represents the solution to the system. By rounding this point to three decimal places as required, you ensure precision in reporting your answer. This method gives a visual appreciation of where the two conditions meet, highlighting how such systems of equations work.