Problem 85 - 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. $$\left\\{\begin{array}{l} 5 b+10 a=20.2 \\ 10 b+30 a=50.1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(a = 0.97\) and \(b = 2.1\).

Step 1: Multiply the Equations

In order to perform the elimination method, we first need to make the coefficients of \(a\) or \(b\) in both equations the same. Since the coefficient of \(a\) in the second equation is 3 times the coefficient of \(a\) in the first equation, multiply the first equation by 3: \[15b + 30a = 60.6\]. Now our system of equations looks like this: \[\left\{\begin{array}{l} 15b+30a=60.6 \ 10b+30a=50.1\end{array}\right.\]

Step 2: Subtract Equations

With the coefficients of \(a\) matching in both equations, we can now subtract the two equations in order to eliminate \(a\). This yields:\[5b = 10.5.\]

Step 3: Solve for \(b\)

Next, divide both sides of the equation by 5 to solve for \(b\):\[b = 2.1.\]

Step 4: Substitute the Value of \(b\) Into One of the Original Equations

We can choose the first original equation to substitute the value of \(b\) into because it has smaller coefficients. Here's what that looks like: \[5(2.1) + 10a = 20.2\].

Step 5: Solve for \(a\)

In this final step, simplify the equation and solve for \(a\):\[10.5 + 10a = 20.2\]\[10a = 20.2 - 10.5\]\[10a = 9.7\]\[a = 0.97.\]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations that share the same set of unknowns. In this exercise, the system includes two equations with two variables, namely \(a\) and \(b\).
The system of equations given is:

• \(5b + 10a = 20.2\)
• \(10b + 30a = 50.1\)

The goal is to determine the values of \(a\) and \(b\) that satisfy both equations simultaneously. Solving a system of equations can be done using various methods, such as substitution, elimination, or graphing. Each method has its own advantages depending on the specific system given.

Elimination Method

The elimination method is a technique used to solve systems of equations. The main idea is to eliminate one of the variables by adding or subtracting the equations. In this exercise, notice that by manipulating the equations, we can make the coefficient of one variable the same in both equations.

Let's take a step-by-step look at the elimination method used here:

• First, the first equation is multiplied by 3 to align the coefficients of \(a\).
• This gives us a new system: \[ \begin{align*} 15b + 30a & = 60.6 \ 10b + 30a & = 50.1 \end{align*} \]
• Next, the second equation is subtracted from the first to eliminate \(a\), resulting in \(5b = 10.5\).
• Finally, by solving this equation, we find \(b = 2.1\).

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or computer software, to visually represent equations or data. When dealing with systems of equations, graphing utilities can:

• Graph each equation and visibly show where they intersect.
• Provide an intuitive understanding of the solution.
• Allow for systematic confirmation of algebraic solutions.

To verify solutions derived algebraically, you can plot each equation using a graphing utility. In this case, graphing is used to confirm that the intersection point of the two lines corresponds to the values \(a = 0.97\) and \(b = 2.1\). This intersection represents the solution to the system of equations.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable(s) that make the equation true. In linear systems, each equation represents a line, and the solution is where these lines intersect.For the equation resulting from our elimination process:\[5b = 10.5\]The solution is straightforward. Divide both sides by 5:\[b = 2.1\]Substituting \(b = 2.1\) into the original equation \(5b + 10a = 20.2\), gives:\[5(2.1) + 10a = 20.2\]Simplifying the equation leads to:

• \(10.5 + 10a = 20.2\)
• \(10a = 20.2 - 10.5 = 9.7\)
• \(a = 0.97\)

This clear step-by-step process demonstrates how each equation is solved to find the unknown values, completing the solution of the system.