Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousand. $$\begin{aligned}&\frac{11}{3} x+y=0.5\\\&0.6 x-y=3\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system is approximately (x, y) = (-0.782, -2.531).

Step 1: Enter the equations into the graphing calculator

To solve the system using a graphing calculator, input the equations into the calculator. For the first equation, rewrite it as \( y = -\frac{11}{3}x + 0.5 \). The second equation can be simplified to \( y = 0.6x - 3 \). Enter these two functions into the calculator.

Step 2: Plot the graphs of both equations

Use the graphing feature to plot the graphs of \( y = -\frac{11}{3}x + 0.5 \) and \( y = 0.6x - 3 \). Ensure that both lines are visible on a single graph screen by adjusting the window settings if necessary.

Step 3: Analyze the intersection point

Observe the graph to locate the point where the two lines intersect. This point represents the solution of the system as it satisfies both equations simultaneously. Use the 'Calculate' feature, often found in the 'Intersect' menu of the calculator, to find the exact coordinates of the intersection point.

Step 4: Round the solution to the nearest thousandth

Once the calculator provides the intersection point, round the x and y coordinates to the nearest thousandth. For example, if the coordinates are (x = 0.35, y = -0.5), check if any digit beyond the third decimal place affects the rounding of the third decimal place.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal when solving these systems is to find the set of values for the variables that satisfy all the equations simultaneously. In other words, you are looking for a common solution for each of the equations involved.

There are several methods to solve a system of equations, including:

• Substitution Method: Solving for one variable in terms of others and substituting it back.
• Elimination Method: Adding or subtracting equations to eliminate a variable.
• Graphical Method: Plotting each equation on the graph and finding the point(s) where they intersect.

In this exercise, we focus on using a graphing calculator to find the solution graphically.

Intersection Point

The intersection point is the point where the graphs of the equations in the system meet. This point holds the values that satisfy each equation, making it the solution to the system of equations.

Finding the intersection point graphically can be done by:

• Graphing each equation as a line in the coordinate plane.
• Observing where the lines cross each other.
• Using the graphing calculator's 'Intersect' feature to pinpoint the exact coordinates of the intersection.

By identifying the intersection point correctly, you can effectively solve problems involving a system of linear equations.

Graphical Solution

A graphical solution involves plotting the equations on a graph and visually identifying the solution to the system. This method is particularly useful because it provides a visual representation of where the equations intersect, offering an immediate insight into the nature of the solution.

To achieve a graphical solution using a graphing calculator, you should:

• Input the equations into the calculator.
• Graph the equations to view them on the same coordinate plane.
• Adjust the viewing window if necessary to ensure visibility of the intersection point.
• Use graphical tools within the calculator to uncover the precise intersection coordinates.

Linear Equations

Linear equations are equations of the first order and graph as straight lines when plotted on a coordinate plane. They generally take the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Key characteristics of linear equations include:

• They have constant slopes, which means they never curve.
• The solutions can be easily visualized as points on the line.
• Multiple linear equations can be used together to form a system, where finding the intersection often requires identifying where their graphs meet.

Understanding linear equations is crucial for solving many algebraic problems and can be particularly helpful in applying graphical solutions to systems of equations.