Problem 49 - 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. $$ \begin{aligned} 0.15 x+0.27 y &=0.39 \\\\-0.34 x+0.56 y &=1.8 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x \approx 1.17\) and \(y \approx 0.78\).

Step 1: Graph the Equations

Enter the equations into a graphing device. Start with the equation \(y_1 = \frac{0.39 - 0.15x}{0.27}\) for the first equation, and \(y_2 = \frac{1.8 + 0.34x}{0.56}\) for the second equation. Ensure both equations are set up as functions of \(y\).

Step 2: Plot the Intersection

Utilize the graphing device's intersect function to find and plot the point where the two graphs intersect. Zoom in or adjust scales if necessary to clearly view the intersection point.

Step 3: Identify Intersection Coordinates

Once you have plotted the intersection, read the coordinates of this point from the graph. These coordinates represent the values of \(x\) and \(y\) that solve the system of equations.

Step 4: Round the Solution

Round the \(x\) and \(y\) coordinates of the intersection point to the nearest hundredth to find your solution for the system of equations.

Key Concepts

Intersection Point

In solving systems of equations, finding the intersection point is crucial. It is the point where the graphs of the equations meet. This point represents the solution to the system of equations. In the context of linear equations, this intersection point shows the values of the variables that make both equations true simultaneously.
Think of two lines on a graph. Each represents a different equation. Where they cross is the intersection point. In algebraic terms, this means substituting these coordinate values into both equations should satisfy them.

• Both equations should be converted to the form of a line: typically in "y = mx + b".
• Graph the lines to find where they cross.
• The intersection point's coordinates, usually noted as \((x, y)\), give us the solution to the system.

Graphing Calculator

A graphing calculator is an electronic tool that plots graphs, solves equations, and performs various other math tasks. For solving linear equations, such as this exercise, a graphing calculator is invaluable as it quickly displays the graphs of the equations.
When using a graphing calculator to find the intersection point:

• First, input the equations in function form, like \(y_1\) and \(y_2\).
• The calculator will then show each equation as a line on a graph.
• You can use the calculator's "intersect" or equivalent function to pinpoint where the two lines meet, showing the solution to the system.

Utilizing a graphing calculator not only saves time but also increases accuracy when dealing with systems of equations.

Linear Equations

Linear equations are equations of the first degree, meaning their variables are to the power of one. They graph as straight lines.
When dealing with linear equations in a system:

• Each equation is rewritten in a slope-intercept form \(y = mx + b\).
• The slope \(m\) represents how steep the line is, while the intercept \(b\) shows where the line crosses the y-axis.
• Systems of linear equations can have one solution (the lines intersect once), no solution (the lines are parallel), or infinitely many solutions (the lines are the same).

This exercise focuses on finding the one solution – where these lines intersect on a graph, indicating consistent and independent equations.

Solution Rounding

When solving systems of equations with a graph, it’s common to obtain decimal values for the solution. Proper rounding is important so that the answers align with the required precision.
In math, rounding improves clarity by adjusting numbers to the nearest desired place value. Here's why this is significant:

• Ensures precision and meets specified tolerance, like rounding to the nearest hundredth in this exercise.
• Aids in real-world applications where exact numbers aren't always practical but rounded figures work better.
• Use the rounding standard: if the digit after the place you are rounding to is 5 or greater, round up. Otherwise, round down.

For example, a value of \(x = 1.234\) rounds to \(x = 1.23\) when rounded to the nearest hundredth. It ensures the presentation of solutions is both professional and precise.