Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Write the system of equations represented by the augmented matrix. Use backsubstitution to find the solution. (Use \(x, y, z\), and \(w .)\) $$ \left[\begin{array}{rrrr} \mathbf{1} & \mathbf{3} & -\mathbf{1} & 15 \\ \mathbf{0} & \mathbf{1} & \mathbf{4} & -12 \\ 0 & 0 & 1 & -5 \end{array}\right] $$

Step-by-Step Solution

Answer

After using backsubstitution, the solution to your system of equations is \(x = -4\), \(y = 8\), and \(z = -5\). The variable \(w\) is not included in the matrix, therefore it is considered as a free variable.

Step 1: Observe the Matrix

There are four columns in the augmented matrix. The first three columns represent coefficients of the variables \(x, y, z\) and the last column represents the constant terms. Hence, the first step is to use the matrix to form the system of equations.

Step 2: Form the System of Equations

Using the matrix, each row is translated into these equations: \(1x + 3y - 1z = 15\) \(0x + 1y + 4z = -12\) \(0x + 0y + 1z = -5\) Therefore, the system of equations becomes: \(x + 3y - z = 15\) \(y + 4z = -12\) \(z = -5\)

Step 3: Use Backsubstitution

The third equation gives the value of \(z = -5\). Substituting \(z = -5\) into the second equation gives \(y + 4(-5) = -12\) which simplifies to \(y = -12 + 20 = 8\). Similarly, inputting \(z = -5\) and \(y = 8\) into the first equation gives \(x + 3(8) - (-5) = 15\), which simplifies to \(x = 15 - 24 + 5 = -4\).

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations that share common variables. For a system involving variables like \(x\), \(y\), and \(z\), an equation could look like \(a_1x + b_1y + c_1z = d_1\). The goal is to find values for these variables that satisfy all the equations at once.
A real-world example involves calculating the price of groceries with differing prices per item. This can be translated into mathematical equations, forming a system. In our example, the matrix given is an augmented matrix, representing a system with three equations involving variables \(x\), \(y\), and \(z\).
To express this matrix in equation form, each row becomes an equation:

• The first row \(1x + 3y - 1z = 15\)
• The second row \(y + 4z = -12\)
• The third row \(z = -5\)

By translating a matrix into equations, you clearly see the relationships between variables, making it easier to find the solution.

Backsubstitution Technique

Backsubstitution is a method used to solve a system of equations once it has been put in a manageable form. It is most effective when you have a triangular form of the matrix or system, where equations are ordered in such a way that you can solve for one variable at a time.
In our case, we start with the simplest equation, \(z = -5\). Knowing \(z\) makes it easy to substitute back into the previous equations:

• First, substitute \(z = -5\) into the second equation \(y + 4z = -12\), to solve for \(y\).
• Then, knowing \(y = 8\) and \(z = -5\), substitute both back into the first equation to solve for \(x\).

Backsubstitution provides a clear path for resolving systems of equations, especially when direct solutions are not possible.

Solving Equations Step by Step

To accurately solve a system of equations, it's crucial to approach step by step. Each step builds on the results from the previous one, ensuring that the deductions and substitutions are accurate.
Here's the approach to solving the system with our example:

• Identify the system of equations: Translate the augmented matrix into its corresponding equations: \(x + 3y - z = 15\), \(y + 4z = -12\), \(z = -5\).
• Start from the simplest equation: Solve for \(z\) to get \(z = -5\). This is straightforward and gives a concrete value.
• Substitute back: Use the value of \(z\) in the second equation to find \(y = 8\).
• Continue substituting: Finally, substitute both \(y\) and \(z\) into the first equation to find \(x = -4\).

By following these steps one by one, you can methodically find the correct values for all variables, building your solution on a solid understanding of each operation.