Problem 56 - 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. $$\left\\{\begin{array}{c} \frac{3}{4} x-\frac{5}{2} y=-9 \\ -x+6 y=28 \end{array}\right.$$

Step-by-Step Solution

Answer

The exact solution may vary due to approximation. It will be the coordinates (x, y) at the intersection of the graphs of the two equations, rounded to three decimal places.

Step 1: Graph The First Equation

Graph \( \frac{3}{4} x-\frac{5}{2} y=-9 \). With the most graphing utilities, this can be achieved by inputting the equation just as it is and the graph will be displayed.

Step 2: Graph The Second Equation

Graph \( -x+6 y=28 \). This can be done in a similar way as the first equation by inputting the equation into the graphing utility and displaying the graph. The graphs of the two equations should be viewed on the same set of axes to be able to find the intersection.

Step 3: Find The Intersection

Observe the graph to find the intersection point of the two lines. This point represents the solution to the system of the equations. When the lines intersect, it means the x, y coordinates at that point satisfy both equations.

Step 4: Round Results

Since the problem asks for the solution to be rounded to three decimal places, make sure to approximate the coordinates x and y to three decimal places.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is fundamental in algebra and helps students visualize solutions to a system of equations. A linear equation can be written in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To graph a linear equation:

• Identify the y-intercept \( b \), where the line crosses the y-axis.
• Use the slope \( m \) to find another point, moving up/down and left/right from the y-intercept depending on the sign and magnitude of the slope.
• Draw a straight line through these points extending in both directions.

For more complex equations, like \( \frac{3}{4}x - \frac{5}{2}y = -9 \), rearrange the equation to solve for y first and then follow the same steps. Graphing utilities automate this by plotting points and drawing the line based on the input equation.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution to the system is the set of values that satisfies all equations simultaneously. Systems of linear equations can have:

• A single point of intersection, representing one unique solution.
• No point of intersection (parallel lines), representing no solution.
• An infinite number of solutions if the lines coincide (same line).

Graphically, we look for the point where the lines representing each equation meet. In the given exercise, the equations \( \frac{3}{4}x - \frac{5}{2}y = -9 \) and \( -x + 6y = 28 \) will intersect at a point that is the solution to both equations.

Approximating Solutions

When exact solutions are unobtainable through algebraic methods, approximating solutions becomes a practical approach. Using a graphing tool:

• Plot the lines as described in previous steps.
• Zoom in on the point of intersection for accuracy.
• Estimate the coordinates to the desired level of precision.

For our example, after plotting \( \frac{3}{4}x - \frac{5}{2}y = -9 \) and \( -x + 6y = 28 \), the intersection must be determined visually and the coordinates approximated to three decimal places as per the problem's instructions.

Graphing Utility Usage

Graphing utilities, whether online platforms, mobile apps, or graphing calculators, offer an easy way to visualize mathematical concepts, especially for systems of equations. To effectively utilize a graphing utility:

• Input the equation exactly as given, ensuring that the calculator is set to the correct mode (e.g., radians or degrees for trigonometric functions).
• Adjust the viewing window to include the relevant parts of the graph, especially the intersection point for systems of equations.
• Use features like 'trace' or 'zoom' to accurately approximate the solution.

Incorporate stepwise use of these functions to arrive at a solution, as done with the equations provided in our exercise. Such tools are particularly useful in visually determining where two lines meet, which is the solution to the system of equations.