Problem 48 - College Algebra

Question

Use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.5 x+0.3 y &=4 \\ 0.25 x-0.9 y &=0.46 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system is approximately (5.12, 3.85).

Step 1: Graph the First Equation

Begin by rewriting the first equation in slope-intercept form (\(y=mx+c\)). The given equation is \(0.5x + 0.3y = 4\). Solving for \(y\), we subtract \(0.5x\) from both sides, then divide by 0.3: \(y = -\frac{0.5}{0.3}x + \frac{4}{0.3}\). Graph this equation on a graphing device.

Step 2: Graph the Second Equation

Now, take the second equation \(0.25x - 0.9y = 0.46\) and write it in slope-intercept form. Solve for \(y\): subtract \(0.25x\) from both sides and divide by -0.9: \(y = \frac{0.25}{0.9}x - \frac{0.46}{0.9}\). Graph this equation as well on the same axes as the first equation.

Step 3: Use the Intersect Function

Use the intersect feature on the graphing device to find the intersection point of the two lines. This point represents the solution \((x, y)\) to the system of equations.

Step 4: Record the Intersection Point

Once the graphing device provides the intersection point, round the \(x\) and \(y\) coordinates to the nearest hundredth as specified. For instance, if the intersection point is (5.123, 3.849), you would record it as (5.12, 3.85).

Key Concepts

Graphing Calculator

When dealing with systems of linear equations, a graphing calculator becomes an invaluable tool. Graphing calculators allow you to visualize equations, making it easier to find solutions. For this exercise, you need to graph two equations and find where they intersect. A graphing calculator helps you plot each equation precisely on the same graph.

• Start by entering each equation into the calculator. You'll usually need to put them in the slope-intercept form first, which simplifies the graphing process.
• Once both equations are in, the graphing calculator will display two lines on a coordinate plane.
• You can then use the intersect feature to determine where these two lines meet, or cross each other.

This tool simplifies solving complex systems, especially when dealing with decimals or equations that aren't easily handled by hand.

Slope-Intercept Form

Understanding the slope-intercept form is central in graphically solving systems of equations. This form, given by \( y = mx + b \), includes two key components:

• \(m\): The slope, which indicates the steepness and direction of the line. A positive slope means the line ascends, while a negative slope means it descends.
• \(b\): The y-intercept, which is where the line crosses the y-axis.

For this exercise, rewrite each given equation into this format to make graphing easier. Having them in slope-intercept form allows you to quickly determine the line's behavior just by looking at its equation, making it strategically beneficial before plotting the line.

Intersection Point

The intersection point is the primary objective when solving systems of equations graphically. This is the point on the graph where two lines cross, representing the solution \((x, y)\) to the system. Finding the intersection point requires the following:

• Graph both equations on the same coordinate plane using your graphing calculator.
• Observe where the lines intersect; the point at which this occurs provides values for \(x\) and \(y\) that satisfy both equations simultaneously.
• Use the calculator's intersect function, which is designed to accurately calculate this point and provide precise values for further analysis.

Being able to identify this point on the graph not only gives you the solution but also offers visual confirmation that the equations are correctly balanced and intersect as expected.

Rounding Numbers

Rounding numbers is an essential skill in graphing calculations, ensuring precision while maintaining simplicity. When you derive the intersection point from a graphing calculator, it's usually accurate to many decimal places. However, for practical purposes, rounding is often required. In this exercise, you are asked to round to the nearest hundredth.
Here's how to do it:

• Identify the digit in the hundredth place (second digit after the decimal).
• Look at the digit immediately following. If it's 5 or greater, round up by adding one to the hundredth place. Otherwise, keep the hundredth place as is.
• For instance, if your result was \((5.123, 3.849)\), rounding would adjust this to \((5.12, 3.85)\).

Rounding ensures that your answers are easier to read and compare, without sacrificing too much accuracy. It also keeps results consistent, especially when presenting data or reporting findings.