Problem 48 - 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. $$ \begin{aligned} 0.5 x+0.3 y &=4 \\ 0.25 x-0.9 y &=0.46 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 5.75\), \(y = -1.50\).

Step 1: Graph the equations

First, enter the equation \(0.5x + 0.3y = 4\) and \(0.25x - 0.9y = 0.46\) into a graphing calculator or software. Plot both lines on the same graph.

Step 2: Identify points of intersection

Look at the graph you've plotted. Locate the point where both lines intersect. This point represents the solution to the system of equations.

Step 3: Use the intersect function

Utilize the intersect feature on your graphing device to find the exact coordinates of the intersection point. The intersect function will help to accurately calculate this point.

Step 4: Round the solution

Once the coordinates of the intersection are determined using the intersect function, round the values of \(x\) and \(y\) to the nearest hundredth to get the final solution.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same variables. Solving a system means finding the values of these variables that satisfy all equations simultaneously. In our exercise, we are dealing with two linear equations:

• \(0.5x + 0.3y = 4\)
• \(0.25x - 0.9y = 0.46\)

When dealing with such systems, different methods can be used, such as substitution, elimination, or graphical methods. Utilizing a graphing calculator is a powerful graphical method, especially for visual learners, because it allows you to see where the lines representing the equations intersect, offering a tangible visualization of the solution.

Intersection Point

An intersection point is the location in the coordinate system where two lines meet. In the context of a system of equations, this point provides the solution, telling us the values of \(x\) and \(y\) that satisfy both equations. Graphically, it is the coordinate on the plane where the two lines cross.

To find the intersection point accurately with a graphing calculator, you input both equations and use its intersect function. This function calculates the exact coordinates of this crossing point, which represents the solution to our system of equations. Locating this point is crucial, as it translates the visual graph data into precise numerical solutions.

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra and geometry to describe geometric figures using a coordinate system. Here, we use the Cartesian coordinate system, which allows us to express lines and curves algebraically, as with our given equations.

This form relies on visual plotting, where each solution is represented as an intersection on the graph of the coordinate plane. Through coordinate geometry, we can interpret the system of equations as two straight lines, and their intersection as a solution showing clear geometric insight into algebraic relationships. Successfully plotting these lines on a graphing device helps in representing the system visually, making it easier to understand.

Rounding Numbers

Rounding numbers is a mathematical approach to simplifying numbers for clarity and convenience. In our context, after finding the intersection point of the system of equations, the exact coordinates might have numerous decimal places. Rounding these coordinates to the nearest hundredth is required to make the result more manageable.

For example, if upon calculating, the intersection point is found to be \( (2.3447, -1.6789) \), rounding entails adjusting these to two decimal points, resulting in \( (2.34, -1.68) \). This method ensures precision and uniformity in reporting answers, maintaining a standard for accuracy while being practical for further computations.