Problem 45 - Intermediate Algebra

Question

Use your graphing calculator to help determine the solution set for each of the following systems. Be sure to check your answers. (a) \(\left(\begin{array}{l}3 x-y=30 \\ 5 x-y=46\end{array}\right)\) (b) \(\left(\begin{array}{l}1.2 x+3.4 y=25.4 \\ 3.7 x-2.3 y=14.4\end{array}\right)\) (c) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-3 y=10 \\ 3 x+5 y=53\end{array}\right)\) (e) \(\left(\begin{array}{l}4 x-7 y=-49 \\ 6 x+9 y=219\end{array}\right)\) (f) \(\left(\begin{array}{l}3.7 x-2.9 y=-14.3 \\ 1.6 x+4.7 y=-30\end{array}\right)\)

Step-by-Step Solution

Answer

Use the graphing calculator to find intersections for each system to solve for \(x\) and \(y\).

Step 1: Understand the Problem

We have a system of linear equations that we need to solve. The task is to use a graphing calculator to find the solution set, which means the values of \(x\) and \(y\) that satisfy both equations in each system.

Step 2: Enter the Equations in the Calculator

Use a graphing calculator's equation or graphing function tool. For each system, enter the two equations separately. For example, enter `3x - y = 30` and `5x - y = 46` for part (a) into the calculator.

Step 3: Graph the Equations

Plot both equations on the graphing calculator. The intersection point of the two lines represents the solution to the system of equations. Look at the graph to visually find the intersection point, which gives the values of \(x\) and \(y\).

Step 4: Find the Intersection

Use the 'intersect' function on the calculator to find the exact values of \(x\) and \(y\) at the intersection point. For each part, record the intersection coordinates as the solution set.

Step 5: Verify the Solution

Substitute the values of \(x\) and \(y\) obtained from the intersection into both original equations to ensure they hold true. This step confirms that the solution is accurate.

Step 6: Complete for Each System

Repeat Steps 2 through 5 for all the given systems (b through f). Enter the corresponding equations, find graphs, set intersections, and verify the solutions.

Key Concepts

Graphing Calculator

Graphing calculators are incredibly useful tools for solving systems of linear equations. These calculators allow you to visually represent equations, which can simplify the process of finding solutions. To use a graphing calculator effectively in this context, follow these steps:

• Enter each equation of the system into the calculator's equation function.
• Access the graphing function to plot these equations on the coordinate system.
• Look for where the plotted lines intersect, as this is where you'll find your solution.
• Often, graphing calculators have an 'intersect' feature to pinpoint the exact values of the intersection point.

By relying on the visual and computational capabilities of a graphing calculator, you can quickly and accurately find the solution to the system of equations.

Linear Equations

Linear equations represent straight lines when graphed on a coordinate plane. They are usually expressed in the form of \(ax + by = c\). Key characteristics of linear equations include:

• Consistency in the change of values between the variables.
• You are dealing with a plain surface when visualizing multiple equations together.

In a system of linear equations, you are comparing two or more linear relationships and seeking a common solution. Each equation represents constraints, and usually, the solution to these constraints is found at the intersection of their lines. Understanding that these equations form lines helps in predicting their behavior and the possible number of solutions, be it one, many, or none.

Intersection Point

The intersection point in a graph of linear equations is crucial. It is where the lines that represent the equations meet. This point satisfies both equations simultaneously, making it the solution to the system.

• To find an intersection point, plot each equation on the graph.
• Look for the coordinates where the lines cross.
• The coordinates give you the specific values of \(x\) and \(y\) that work for both equations.

By using a graphing calculator's intersect function, you can easily compute these values. Identifying the intersection is what turns the process from a purely theoretical exercise into a practical method of problem-solving by directly observing where constraints meet.

Solution Verification

Verification is a crucial step when solving systems of equations to ensure the solution is correct. After finding the intersection point, substitute the \(x\) and \(y\) values back into the original equations.

• Replace the \(x\) and \(y\) in both equations to see if each equation holds true.
• If both equations are satisfied, you have verified your solution is accurate.
• If either equation doesn't hold, you'll need to re-evaluate the intersection point.

Verification confirms that the solution not only mathematically fits but also meets the requirements of both equations in the system. By double-checking your solution, you'll avoid mistakes and gain confidence in the accuracy of your answers.