Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 7 b+21 a &=13.1 \\ 21 b+91 a &=-2.8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution for the system of equations yields \(a = -0.1265\) (slope) and \(b = 1.904\) (y-intercept). Therefore, the least squares regression line is \(y = -0.1265x + 1.904\).

Step 1: Rewrite the system of equations in matrix form

Rewrite the linear system in matrix form. The system of equations can be represented as \(A \cdot x = b\), where \(A\) is matrix of coefficients, \(x\) is the vector of variables, and \(b\) is the vector of constants. In this case,\n\[\left[\begin{array}{cc}7 & 21 \ 21 & 91\end{array}\right]\left[\begin{array}{c}b \ a\end{array}\right]=\left[\begin{array}{r}13.1 \ -2.8\end{array}\right].\]

Step 2: Solve for the variables

Solve for \(b\) and \(a\) using Gaussian elimination, Cramer's rule, or another method suitable for solving system of linear equations in two variables. For simplicity, we can use a graphic calculator or an online math tool to solve the system. The solution yields: \(a\) \approx -0.1265, \(b\) \approx 1.904.

Step 3: Interpret the results

The variables \(a\) and \(b\) found represent the slope and the y-intercept of the least squares regression line, respectively. Hence, the equation of the line is \(y = ax + b\), or in this case \(y = -0.1265x + 1.904\).

Key Concepts

Matrix Form

The matrix form of a system of equations is a concise way of representing and solving linear equations. In our exercise, we have two equations with two unknowns. We can express these equations in a matrix form:

• Let matrix **A** consist of the coefficients of the variables: \[\left[\begin{array}{cc}7 & 21 \ 21 & 91\end{array}\right]\]
• Vector **x**, which contains the variables we want to solve for, is: \[\left[\begin{array}{c}b \ a\end{array}\right]\]
• Finally, vector **b** includes the constants from the right side of the equations: \[\left[\begin{array}{r}13.1 \ -2.8\end{array}\right]\]

Representing the system in this form helps us visualize and set up problems for computational solutions more effectively. It highlights the structure of the problem, making it easier to apply solution techniques like Gaussian Elimination.

System of Linear Equations

Understanding a system of linear equations is key in solving problems involving multiple variables. Here, we have:

• Equation 1: \(7b + 21a = 13.1\)
• Equation 2: \(21b + 91a = -2.8\)

The challenge is to determine the values of \(b\) and \(a\) that satisfy both equations simultaneously. These equations graph as straight lines, and the solution is the point where these lines intersect.
The beauty of a linear system is that, given consistent equations, there exists a unique or infinite set of values for the variables involved. Systematic approaches like substitution, elimination, and matrix operations can be used to find these solutions.

Gaussian Elimination

Gaussian Elimination is a step-by-step procedure used to solve systems of linear equations. It involves:

• Transforming the system's matrix to the upper triangular form, also called row-echelon form. This makes the system easier to solve.
• Performing back substitution, where we solve for variables starting from the bottom equation back to the top.

In the exercise, solving the system by Gaussian Elimination involves switching the equations to isolate one variable:

• Transform the first row so one element becomes a pivot (usually make the first element 1 by dividing the row).
• Eliminate the variables below the pivot by subtracting suitable multiples of the pivot row from the rows below.
• Once in upper triangular form, solve starting from the bottom equation, progressing upward.

Using Gaussian Elimination provides a structured method to find the unique solution \(a = -0.1265\) and \(b = 1.904\).

Graphing Utility

A graphing utility is a powerful tool used to visualize and confirm the results obtained from solving equations analytically. Once the least squares regression line is determined by solving the system, it can be plotted using a graphing calculator or a software like Desmos or GeoGebra.
Here's how a graphing utility can help:

• By graphing the line \(y = -0.1265x + 1.904\), you can observe how well it fits the data points given in the problem.
• It provides a visual interpretation of the slope \(a\) and y-intercept \(b\) derived from the calculations. Slope represents how steep the line is, and the y-intercept is where the line crosses the y-axis.
• Adjustments can be made if necessary to fine-tune the visual fit, which is especially helpful if the exercise involves experimental or real-life data.

The graphical representation serves as a confirmation that the calculations yield a line providing a good estimate of the trend among given data points.