Question

You and a friend solve the following system of equations independently. \(\left\\{\begin{array}{rr}2 x-4 y-3 z= & 3 \\ x+3 y+z= & -1 \\ 5 x+y-2 z= & 2\end{array}\right.\) You write your solution set as \((a,-a, 2 a-1)\) where \(a\) is any real number. Your friend's solution set is \(\left(\frac{1}{2} b+\frac{1}{2},-\frac{1}{2} b-\frac{1}{2}, b\right)\) where \(b\) is any real number. Are you both correct? Explain. If you let \(a=3\), what value of \(b\) must be selected so that you both have the same ordered triple?

Short answer

After verification, we find that both solutions are correct as they satisfy the given system of equations. For 'a' equal to 3, solving the resulting equations suggests that the value of 'b' that yields the same ordered triple in both solution sets is 1. Thus, when 'a' equals 3, 'b' must equal 1 for both solution sets to be equivalent.

Step-by-step solution

Verifying the first solution

Substitute the values from the first solution set, \((a,-a, 2 a-1)\), into the original equations to verify if they hold true. If they do, then the first solution set is correct.

Verifying the second solution

Similarly, substitute the values from the second solution set, \(\left(\frac{1}{2} b+\frac{1}{2},-\frac{1}{2} b-\frac{1}{2}, b\right)\), into the original equations. If these values also satisfy the equations, then the second solution set is correct.

Finding the value of b when a=3

Now, we need to find the value of 'b' such that both solution sets yield the same ordered triple when 'a' is given to be 3. To do this, we will set the expressions for 'x', 'y', and 'z' from both solutions equal to each other, and solve for 'b'.

Key concepts

Linear Algebra

The system of equations in this exercise is part of linear algebra, a branch of mathematics focusing on understanding and solving linear equations and systems of equations. These systems are composed of linear expressions – polynomials of the first order.
Linear algebra is foundational because it provides the tools to solve systems like:

• Two or more equations.
• Several variables.
• Multiple solutions.

A system might have a unique solution, no solution at all, or infinitely many solutions, often presented as parametric equations that describe a line or plane.
This particular system involves solving for three variables: \(x\), \(y\), and \(z\), in terms of a single parameter \(a\) or \(b\), which showcases its parametric nature. This indicates that there are infinitely many solutions, signifying that the equations form planes that intersect in a line, thus aligning with the concept of having multiple solutions in linear algebra.

Solution Set

A solution set represents the collection of all possible solutions to a system of equations. It is often expressed in parametric form if there are infinite solutions. This means solutions are defined by one or more parameters (like \(a\) or \(b\) in our exercise).
For example, the solution set \(a,-a, 2a-1\) suggests each variable in the system can be expressed as a function of the parameter \(a\). Similarly, your friend’s solution \frac{1}{2} b+rac{1}{2},-rac{1}{2} b-rac{1}{2}, b\ illustrates another way to express the solutions in terms of \(b\).
It's essential to understand that even if different parameters are used (\(a\) and \(b\)), both forms can describe the same line of intersection if they hold true for the original equations. Therefore, matching these parametric representations can confirm both solutions lead to the same line, guaranteeing their correctness.

Verification of Solutions

Verification is a critical step to ensure that the proposed solution sets are correct. It involves substituting the parametric expressions for the variables back into the original equations.

• Substitute \(x = a\), \(y = -a\), and \(z = 2a - 1\) into all three equations.
• Confirm each equation holds true, indicating the solution set \((a, -a, 2a - 1)\) satisfies the system.
• Similarly, insert \(x = \frac{1}{2}b + \frac{1}{2}\), \(y = -\frac{1}{2}b - \frac{1}{2}\), and \(z = b\) into the equations.
• Check these satisfy the original system too, validating the correctness of this second solution set.

If they do, as is the case in the exercise, both solution sets are verified to depict the same geometric line in the system's solution space.
Once verification is completed, parameters can be set equal to each other to find congruence, such as when \(a = 3\), determining \(b\) for consistency between both parametric solutions.