Problem 46 - College Algebra

Question

Use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{array}{c} 0.1 x+0.2 y=0.3 \\ -0.3 x+0.5 y=1 \end{array} $$

Step-by-Step Solution

Answer

The solution is approximately \((4.62, 5.77)\).

Step 1: Graph the First Equation

First, let's graph the equation \(0.1x + 0.2y = 0.3\). Solve for \(y\):\[0.2y = -0.1x + 0.3\]\[y = -0.5x + 1.5\]Plot the line with the slope \(-0.5\) and y-intercept \(1.5\) on the graphing device.

Step 2: Graph the Second Equation

Now, graph the second equation \(-0.3x + 0.5y = 1\). Solve for \(y\):\[0.5y = 0.3x + 1\]\[y = 0.6x + 2\]Plot this line with the slope \(0.6\) and y-intercept \(2\) on the same graphing device.

Step 3: Find the Intersection Point

Use the intersect function on your graphing device to find the point where the two lines intersect. The intersection point represents the solution to the system of equations.

Step 4: Round the Solution to the Nearest Hundredth

Once you have the intersection point from your graphing device, round the x and y coordinates to the nearest hundredth to get the final solution to the system.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a way to visually represent solutions to linear equations on a coordinate plane. Each linear equation can be transformed into a straight line, which is defined by its slope and its y-intercept. To graph a linear equation, follow these steps:

• Begin by rearranging the equation into the slope-intercept form, which is given by: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• The slope \(m\) represents how steep the line is. It is calculated as the rise over run, or the change in y divided by the change in x.
• The y-intercept \(b\) is the point where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\).
• Plot the y-intercept on the graph first as a starting point.
• From this point, use the slope to determine another point on the line. For example, a slope of 2 means you go up 2 units for every 1 unit you go to the right.
• Draw the line by connecting the two points with a straight edge, and extend it across the graph.

Graphing linear equations helps in understanding how changes in variables affect their relationships, and sets the stage for finding intersections with other lines, which leads to solving systems of equations.

Intersection of Lines

When graphing two or more linear equations, the point where the lines intersect is an important part of solving systems of linear equations. This intersection point is the set of coordinates where both equations are satisfied simultaneously. To find this point, you can:

• Graph each equation on the same set of axes using the slope and y-intercept of each equation.
• Look for the point where the lines cross each other. This is known as the intersection point.

The intersection point can be determined precisely by using graphing tools, such as a graphing calculator, which often include a function to calculate intersections. For example, if the intersect function of a graphing calculator indicates that the intersection occurs at \((2.5, 3.4)\), then these coordinates are the solution to the system. In this context, each coordinate represents the value of the variables that solves both equations simultaneously.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing and solving equations, especially systems of linear equations. These calculators can plot graphs and perform calculations that simplify the solving process significantly. Using a graphing calculator to solve a system of linear equations involves the following steps:

• Input each linear equation into the calculator. The calculator will typically graph the lines based on the information given.
• Once the equations are plotted, use the intersect function to find points where the graphs intersect. This function uses numerical methods to accurately find the intersection points.
• After locating the intersection, the calculator displays the point’s coordinates. These coordinates are the solution to the system of equations, showing the value of \(x\) and \(y\).
• Whenever necessary, round the intersection coordinates to the nearest hundredth as instructed for precise results.

Graphing calculators enhance the understanding of complex algebraic concepts by providing a visual approach. They also save time and reduce errors compared to manual graph plotting, making them invaluable in mathematical problem-solving.