Problem 77 - Precalculus with Limits

Question

Think About It Are the following two systems of equations equivalent? Give reasons for your answer. $$ \left\\{ \begin{aligned} x + 3 y - z & = 6 \\ 2 x - y + 2 z & = 1 \\ 3 x + 2 y - z & = 2 \end{aligned} \right. $$ $$ \left\\{ \begin{aligned} x + 3 y - z = & 6 \\ - 7 y + 4 z = & 1 \\ - 7 y - 4 z = & \- 16 \end{aligned} \right. $$

Step-by-Step Solution

Answer

The two systems of equations are not equivalent.

Step 1: Compare equations from each system

Take a look at the individual equations in each system. In the first system \(x + 3y - z = 6\), \(2x - y + 2z = 1\), \(3x + 2y - z = 2\), and in the second system \(x + 3y - z = 6\), \(-7y + 4z = 1\), \(-7y - 4z = -16\). As we see, the first equation in both systems is identical. Let's consider the other two equations.

Step 2: Try to transform the equations from the first system to match the second

When trying to transform the second equation in the first system \((2x - y + 2z = 1)\) to match the second equation from the second system \((-7y + 4z = 1)\), or the third equation in the first system \((3x + 2y - z = 2)\) to match the third equation from the second system \((-7y - 4z = -16)\), it becomes evident that no combination of transformations can achieve this match.

Step 3: Conclusion

Since it is impossible to obtain the equations from the second system by transforming the ones in the first system, we can conclude that these two systems are not equivalent. They do not represent the same set of solutions.

Key Concepts

Equivalent Systems

Equivalent systems of equations have the same solution set. This means that while individual equations might look different, they represent the same relationships among variables. For systems to be equivalent, each equation in one system can be transformed to one in the other system without altering the fundamental solutions.

• If changing one system's equations leads precisely to the other's equations, they are equivalent.
• An equivalent transformation might involve adding, subtracting, or multiplying equations by a constant.

In the given problem, the first equations of both systems are identical. This is a promising sign of equivalence, initially. However, for two systems to be truly equivalent, all equations across both systems must be transformable into each other, maintaining the solution set. Here, attempting transformations of the remaining equations reveals differences that lead to differing solutions. Therefore, these systems are not equivalent.

Equation Transformation

Equation transformation involves manipulating the equations in a system while preserving their solutions. This is crucial in solving systems of equations, either to derive equivalents or simplify solving. Equation manipulation includes:

• Adding or subtracting equations to eliminate variables.
• Multiplying or dividing equations by non-zero numbers to adjust coefficients.

Transforming equations should not change the solutions. When analyzing the provided systems, attempts to transform equations from the first system into forms present in the second system proved futile. This transformation process is vital to understand adjustments in equations that maintain or reveal equivalent systems, but here it highlighted differences, not equivalences.

Linear Equations

Linear equations are equations involving variables raised only to the first power. These appear as straight lines or planes when graphed. Solving linear equations often involves finding where such lines intersect, which provides common values for all variables involved. Important aspects of linear equations include:

• They have a constant slope or rate of change.
• They are used extensively in modeling real-world situations where relationships are constant.
• Written generally in the form of "ax + by + cz = d" for three variables.

In the given exercise, both systems comprise linear equations. While the first equations are identical, the failure to transform subsequent equations reveals that they represent distinct relationships. Thus, only the first one corresponds directly to an equivalent solution set, while others diverge due to their linear formulation settings.