Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.1 x+0.2 y &=0.3 \\ -0.3 x+0.5 y &=1 \end{aligned} $$

Step-by-Step Solution

Answer

The intersection, and solution to the system, is approximately \((1.67, 2.52)\).

Step 1: Understand the Problem

You are given a system of linear equations:\[0.1x + 0.2y = 0.3\] and \[-0.3x + 0.5y = 1\]. You need to solve this system using graphing and finding the intersection point.

Step 2: Graph the First Equation

Convert the first equation into slope-intercept form. \[0.1x + 0.2y = 0.3\]. Solving for \(y\), we have \[y = -0.5x + 1.5\]. Plot this equation on the graph.

Step 3: Graph the Second Equation

For the second equation \(-0.3x + 0.5y = 1\), solve for \(y\) to get \[y = 0.6x + 2\]. Plot this equation on the same graph as the first equation.

Step 4: Use the Intersect Function

With both equations graphed, use the intersect tool on the graphing calculator to find the intersection point. This location represents the solution to the system of equations.

Step 5: Round the Intersection Point

The intersect function should provide an accurate intersection point. Round this point to the nearest hundredth for both the \(x\) and \(y\) coordinates.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool used to graph equations and visualize their solutions. When solving systems of linear equations, a graphing calculator can be extremely handy.

• After entering your equations, it can generate visual graphs that represent each equation on the same grid.
• Most graphing calculators come equipped with a function to find intersections, which is crucial for determining where two lines meet.
• This technology simplifies the process by allowing students to see the graphical representation, making it easier to understand the relationships between equations.

Using this device can help students avoid errors in manual calculations and focus more on understanding the results. Just remember that accurate graphing is key to finding the correct solution.

Intersection Point

The intersection point of two lines is the location on a graph where the lines cross each other.

• In the context of a system of equations, this point represents the solution to the equations, providing the values of \(x\) and \(y\) that satisfy both equations simultaneously.
• A graphing calculator can be used to find this point by graphing the equations and employing an intersect function.
• It calculates where the two lines intersect and provides the exact coordinates.

To ensure accuracy, once you find the intersection point, you should verify it by substituting these values back into the original equations. This confirms that they satisfy both equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation so that it is easy to graph. The equation takes the form: \[y = mx + b\]

• Here, \(m\) represents the slope of the line, which indicates the steepness and direction.
• The \(b\) is the y-intercept, the point where the line crosses the y-axis.
• Slope-intercept form allows quick graphing since you can immediately plot the y-intercept and use the slope to find another point.

In solving the system of equations, converting each equation to this form simplifies the graphing process. It allows us to instantly visualize where and how each line will appear on the coordinate plane. Once graphed, it is easier to see where the lines might intersect, aiding in finding the solution to the system.