Problem 38 - Intermediate Algebra

Question

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. $$ \left\\{\begin{array}{l} y=-0.45 x+5 \\ y=5.55 x-13.7 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 3.57, y = 3.39 \).

Step 1: Graph the Equations

Plot the equations on a graphing calculator: the first equation is \( y = -0.45x + 5 \) and the second equation is \( y = 5.55x - 13.7 \). Ensure both lines are visible on the screen simultaneously.

Step 2: Find the Intersection Point

Use the graphing calculator's feature to calculate or trace the point where the two lines intersect. This point represents the solution to the system of equations.

Step 3: Read the Intersection Coordinates

Once the intersection is identified, note the x-coordinate and y-coordinate of this point. These are the solution values for \( x \) and \( y \).

Step 4: Round the Solution

Round the coordinates of the intersection point to the nearest hundredth to provide the final answer for the system of equations.

Key Concepts

Solving Systems of Equations

When solving systems of equations, the goal is to find the set of values for the variables that satisfy all equations involved. In this exercise, we are dealing with two linear equations:

• \( y = -0.45x + 5 \)
• \( y = 5.55x - 13.7 \)

Systems of equations can be solved using various methods, such as substitution, elimination, and graphing. Here, we focus on solving the system graphically using a graphing calculator.
By plotting both equations on the same set of axes, we search for a point where the lines intersect. This intersection represents the solution to the system, meaning it's the point where both equations are true simultaneously.

Intersection Point

The intersection point is a critical concept when graphing systems of equations. It represents the coordinates \((x, y)\) where both equations in the system are satisfied.
In this context, it is the point where both lines graphically intersect on the graphing calculator screen.
The graphing calculator has built-in functionalities to trace or calculate the intersection point. Once located, this point provides the values of \(x\) and \(y\) that solve the system.

• To identify the intersection, visually monitor the graph for where the two lines cross.
• Use the calculator's intersection feature to compute the exact coordinates.
• These coordinates should then be rounded as needed, here to the nearest hundredth.

Recognizing the intersection point on the graph is a key step in visual problem-solving, turning graphical interpretations into mathematical solutions.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in solving equations by visualization. Each equation in a linear system represents a line with a particular slope and y-intercept, appearing as equations like \( y = mx + b \).

• Slope \(m\): This number indicates the line's steepness or tilt.
• Y-intercept \(b\): The point where the line crosses the y-axis when \(x = 0\).

To graph these equations accurately on a graphing calculator:- Enter each equation into the calculator's graphing function.- Adjust the viewing window to ensure both lines are visible.- Use the calculator to display both equations on the same graph simultaneously.Graphing visually demonstrates the relationships between equations, and the cross-section or intersection of lines helps identify solutions. This graphical method is particularly useful for obtaining quick, visual solutions to linear systems.