Problem 47 - College Algebra

Question

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned}-0.01 x+0.12 y &=0.62 \\ 0.15 x+0.20 y &=0.52 \end{aligned} $$

Step-by-Step Solution

Answer

The intersection point is approximately \((4.88, 1.47)\).

Step 1: Identify Equations in the System

We have two equations in the system that need to be solved simultaneously for the variables \(x\) and \(y\). The equations are: 1. \(-0.01x + 0.12y = 0.62\)2. \(0.15x + 0.20y = 0.52\).

Step 2: Graph Each Equation

Use a graphing device to plot both equations on the same coordinate plane. The first equation, \(-0.01x + 0.12y = 0.62\), and the second equation, \(0.15x + 0.20y = 0.52\), should be graphed individually, resulting in two lines.

Step 3: Find the Intersection Point

Using the graphing calculator's intersect function, determine where the two lines cross each other on the coordinate plane. This intersect point represents the solution to the system of equations.

Step 4: Round to the Nearest Hundredth

The intersection point will be in the form \((x, y)\). Round both the \(x\) and \(y\) values to the nearest hundredth to get your final answer.

Key Concepts

Understanding the Graphing Calculator

A graphing calculator is a dynamic tool that can be extremely helpful for solving systems of equations. It allows you to plot equations visually by converting them into lines on a coordinate plane.

This visual representation makes it easier to find solutions or understand the behavior of equations. When you input an equation into a graphing calculator, it takes the linear equation like

• \(-0.01x + 0.12y = 0.62\)
• \(0.15x + 0.20y = 0.52\)

and translates them into lines.

It's almost like drawing a map with roads, where each equation becomes a different path on the map. Once both equations are entered, the graphing calculator can then show where these paths intersect. This is particularly useful because solving a system of equations manually can often involve a lot of algebraic manipulation which can be prone to mistakes.

Grasping the Intersection Point

The intersection point is crucial as it is the solution to a system of equations when graphed. Imagine it as a crossing of two paths, where one line represents each equation.

This point of intersection reveals the values of \(x\) and \(y\) that satisfy both equations simultaneously. Once you graph both equations using a calculator, the lines should cross at some point unless they are parallel, indicating no intersection.

When you use the graphing calculator’s intersect function, it pinpoints this crossing area on the graph. By identifying the intersection point, you find the exact values of \(x\) and \(y\) that solve the system. In a practical sense, this is where the two mathematical "roads" meet, showing the only solution possible for the equations given.

Importance of Rounding to the Nearest Hundredth

After finding the intersection point using your graphing calculator, you get specific \(x\) and \(y\) values. These values often come with decimal points that can be long and tricky.

Rounding to the nearest hundredth simplifies these numbers to two decimal places, making them easier to handle and interpret. For example, if you have a value of 2.3456, rounding it to the nearest hundredth would give you 2.35.

This rounding is essential not only for maintaining clarity but also for aligning with instructional requirements or constraints that specify such precision. So, when you take the intersection point and round those values, they become straightforward and ready for interpretation in terms of the exercise solution.