Problem 82 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\\{\begin{aligned} x+3 y-z &=19 \\ y+6 z &=-18 \\ z &=-4 \end{aligned}\right.\) (b) \(\left\\{\begin{aligned} x-y+3 z &=-15 \\ y-2 z &=14 \\ z &=-4 \end{aligned}\right.\)

Step-by-Step Solution

Answer

By solving both systems of equations using matrices, the solution for each system of equations can be determined. Comparing these solutions will provide the answer whether the two systems yield the same solution.

Step 1: Set up the matrices for both systems

Convert each system of linear equations into matrix form, expressing it as \(Ax = b\), where A is a matrix of coefficients, x is a matrix of variables, and b is a vector of constants. For the system (a): \(A = \begin{pmatrix}1 & 3 & -1 \\ 0 & 1 & 6 \\ 0 & 0 & 1 \end{pmatrix}\), \(x = \begin{pmatrix}x \\ y \\ z \end{pmatrix}\), \(b = \begin{pmatrix}19 \\ -18 \\ -4 \end{pmatrix}\). For the system (b): \(A = \begin{pmatrix}1 & -1 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix}\), \(x = \begin{pmatrix}x \\ y \\ z \end{pmatrix}\), \(b = \begin{pmatrix}-15 \\ 14 \\ -4 \end{pmatrix}\).

Step 2: Solve the matrices

Solve each matrix equation (Ax = b) by using the inverse of matrix A. To solve, multiply both sides of the equation by \(A^{-1}\) to get \(x = A^{-1}b\). For the system (a): \(x = \begin{pmatrix}1 & 3 & -1 \\ 0 & 1 & 6 \\ 0 & 0 & 1 \end{pmatrix}^{-1} \begin{pmatrix}19 \\ -18 \\ -4 \end{pmatrix}\). For the system (b): \(x = \begin{pmatrix}1 & -1 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix}^{-1} \begin{pmatrix}-15 \\ 14 \\ -4 \end{pmatrix}\). It is given that z is -4 in both systems, substituting z in y and x equations to find their values for both the systems.

Step 3: Compare the solutions

The last step is to compare the solutions obtained from both the systems. If the obtained values of x, y, and z are the same for both systems, then they yield the same solution.

Key Concepts

Understanding Matrices in Solving Linear Equations

Matrices are like organized tables of numbers, and they are incredibly useful when working with systems of linear equations. Imagine you have a matrix filled with numbers that represent the coefficients from your linear equations. This is your coefficient matrix, denoted as \( A \).

You also have a matrix for your variables \( x \), which usually looks like a single column, and a matrix for your constants \( b \).

• Coefficient Matrix \( A \): This is the matrix filled with the numbers directly in front of the variables in your linear equations.
• Variable Matrix \( x \): This matrix contains the variables you are solving for, like \( x \), \( y \), and \( z \).
• Constant Matrix \( b \): This matrix holds the constant terms, the numbers on the other side of the equal sign.

To solve the system, you rearrange the equations into a matrix equation format: \( Ax = b \). Then use techniques like finding the inverse of \( A \) to solve for \( x \). This results in "decoding" the matrix into the values of your variables. When successful, you've solved the system using matrix algebra.

Understanding Systems of Equations

A system of linear equations is essentially a set of equations with multiple unknowns. Each equation in the system captures a relationship between the variables. For instance, you might have three variables — \( x \), \( y \), and \( z \) — and several equations telling how they interact.

• Linear Equations: These are equations where each term is either a constant or the product of a constant and a single variable.
• System of Equations: This is just a collection of two or more linear equations involving the same set of variables.

To solve a system, you aim to find values for each variable that satisfy all equations simultaneously. You can solve these systems using substitution, elimination, or by using matrices, like in our exercise. Solving systems of equations is crucial because they model real-world problems and can tell us how different factors relate or depend on each other.

Comparison of Solutions

After solving both systems of equations, you need to compare the results. This involves checking whether the values of \( x \), \( y \), and \( z \) that you found for each system are identical.

• Importance of Comparison: It's essential to ensure that your solution is consistent across possibly rephrased or differently set up systems of equations.
• Steps in Comparison: Check each variable from both systems. If \( x \), \( y \), and \( z \) have the same values (as both systems show \( z = -4 \)), then the systems are equivalent and share the same solution.

This step ensures that we truly understand the relationships between variables, ensuring consistency and confirming the correctness of our matrix solutions. It's particularly insightful in showing that, even if equations look different or are arranged uniquely, they can describe the same scenario.