Problem 54 - Algebra A Combined Function

Question

Use a graphing calculator to solve each system. $$ \left\\{\begin{array}{l} y=3.1 x-16.35 \\ y=-9.7 x+28.45 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((4.3, -3.03)\).

Step 1: Understand the Problem

We are given a system of two linear equations with two variables: \( y = 3.1x - 16.35 \) and \( y = -9.7x + 28.45 \). The task is to find the point \((x, y)\) where these two lines intersect.

Step 2: Input Equations into Graphing Calculator

Enter the first equation \( y = 3.1x - 16.35 \) into your graphing calculator as \( Y_1 = 3.1x - 16.35 \). Then enter the second equation \( y = -9.7x + 28.45 \) as \( Y_2 = -9.7x + 28.45 \). This sets up the calculator to display both lines on a graph.

Step 3: Plot the Graphs

Use the graphing feature to plot both equations on the same coordinate plane. This will visually show you where the two lines intersect.

Step 4: Find the Intersection Point

Use the intersection feature of the graphing calculator to find the precise coordinates where the two lines intersect. This feature calculates the exact values of \(x\) and \(y\) at the point of intersection.

Step 5: Interpret the Results

The graphing calculator shows that the intersection occurs at \(x = 4.3\) and \(y = -3.03\). This means that the solution to the system of equations is \((4.3, -3.03)\).

Key Concepts

System of Equations

A system of equations is simply a collection of two or more equations that you need to solve together. Here, we're working with two equations that involve the same variables, specifically the equations

• \( y = 3.1x - 16.35 \)
• \( y = -9.7x + 28.45 \)

These equations share the variables \( x \) and \( y \). Solving the system involves finding the values of \( x \) and \( y \) that satisfy both equations simultaneously. This ensures that we're finding a common point for the two problems presented in the system.
When these kinds of systems are graphically plotted, the solution corresponds to the point where the lines, represented by the equations, intersect on a graph.

Linear Equations

Linear equations form straight lines when plotted on a coordinate plane. Each equation follows the form \( y = mx + b \), where \( m \) and \( b \) are constants.

• The constant \( m \) represents the slope of the line.
• The constant \( b \) is the y-intercept, where the line crosses the y-axis.

For example, in the equation \( y = 3.1x - 16.35 \), 3.1 is the slope, and -16.35 is the y-intercept.
Both equations in our exercise are linear, making them straightforward to identify and graph on a coordinate plane.
Linear equations help us understand not just where two values meet, but also how they change relative to each other.

Intersection Point

In the context of a system of equations plotted on a graph, the intersection point is where the solutions of the equations meet. This point shows the exact values of \( x \) and \( y \) that satisfy both equations.

• For the system provided, that point is \( (4.3, -3.03) \).

Binding the two lines at an intersection point means both linear equations are balanced with those precise values.
Finding the intersection requires using tools like a graphing calculator that can illustrate these points visually and calculate them numerically.
Understanding this concept is crucial as it simplifies solving systems of equations by providing a clear visual and numerical solution.

Coordinate Plane

The coordinate plane is the space where equations are graphed. It's a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis), like a giant grid.
Each point on this plane corresponds to a pair of numerical values that provide coordinates (\( x, y \)) which describe the location of a point.

• In our exercise, the coordinate plane helps us plot the lines \( y = 3.1x - 16.35 \) and \( y = -9.7x + 28.45 \).
• By plotting these lines, the intersection gives both a visual and numeric solution to the system of equations.

Using a graphing calculator to display these points on the coordinate plane drastically simplifies the process of interpreting results.
This tool provides an intuitive way to solve systems of equations by showing graphically how and where two lines meet.