Problem 39 - Algebra 2

Question

To use a TI-83/84 Plus to solve a system of equations, graph the equations. Then, select INTERSECT, which is option 5 under the CALC menu, to find the coordinates of the point of intersection to the nearest hundredth. \(y=0.125 x-3.005\) \(y=-2.58\)

Step-by-Step Solution

Answer

The point of intersection is approximately (3.40, -2.58).

Step 1: Enter Equations into the Graphing Calculator

Turn on the TI-83/84 Plus and press the 'Y=' button. Enter the first equation, \(y = 0.125x - 3.005\), into \(Y_1\). Then enter the second equation, \(y = -2.58\), into \(Y_2\).

Step 2: Graph the Equations

Press the 'GRAPH' button to display the graphs of the two equations. Observe where they intersect on the calculator's display.

Step 3: Access the CALC Menu

Press '2ND' and then 'TRACE' to open the CALC menu. This menu allows us to calculate various features of the graph, including intersections.

Step 4: Select the INTERSECT Option

In the CALC menu, choose option 5 which is 'INTERSECT'. This function will help us find the coordinates of the intersection point of the two graphs.

Step 5: Identify the Intersection Point

The calculator will ask three prompts: 'First curve?', 'Second curve?', and 'Guess?'. For each prompt, ensure you select the curves you have graphed, then press 'ENTER'. The calculator will then compute and display the intersection point.

Step 6: Record the Intersection Coordinates

Once the calculator shows the intersection point, write down the coordinates as displayed to the nearest hundredth. For this problem, the coordinates should be approximately (3.40, -2.58).

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. Solving a system means finding the values of the variables that satisfy all equations simultaneously. For example, the system given in the problem is:\[\begin{align*}y &= 0.125x - 3.005 \y &= -2.58\end{align*}\]Here, both equations are linear, which means they graph as straight lines. An important step when solving these systems is finding a solution that fits both equations. This is typically represented as the intersection point on a graph.

Intersection Point

The intersection point is where two graphs meet on a coordinate plane. This point represents the set of variable values that satisfy both equations in a system. When graphing two equations, the intersection point is crucial because it indicates where both equations hold true. In this specific exercise, we’re seeking the intersection of a linear function and a horizontal line.

• The line equation: \(y = 0.125x - 3.005\)
• The horizontal line: \(y = -2.58\)

By using a graphing calculator, like the TI-83/84 Plus, you can visually determine this point quickly. The coordinates of this intersection give you the solution to the system of equations.

TI-83/84 Plus

The TI-83/84 Plus is a powerful tool for solving systems of equations graphically. It's a series of graphing calculators that can perform a wide variety of mathematical operations. To solve systems of equations using a TI-83/84 Plus, follow these steps:
1. **Turn on the calculator.** Start by pressing the 'Y=' key to enter the equations screen.
2. **Input the equations:** Enter the equations one by one into the fields provided (e.g., \(Y_1\) and \(Y_2\)).
3. **Graph the equations:** Press 'GRAPH' to visualize them.
4. **Find the intersection:** Use '2ND' + 'TRACE' to access the CALC menu and select 'INTERSECT.'
The calculator then provides options to identify the intersection point, a critical part of solving the system.

Graphing Equations

Graphing equations allows you to visually explore the relationships between variables. Graphing helps not only in understanding the behavior of functions but also in solving systems of equations.
- **Linear equations** like \(y = 0.125x - 3.005\) form straight lines.- **Horizontal lines** such as \(y = -2.58\) stretch parallel to the x-axis.
Utilizing a graphing calculator, like the TI-83/84 Plus, offers a convenient way to draw these graphs simultaneously. This visual approach is especially useful for identifying the intersection point.
Once both equations are graphed, the intersection can clearly show the solution, making complex mathematical concepts more manageable and easier to understand for students.