Problem 12 - Precalculus

Question

In Exercises \(9-26\), put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent. $$ \left\\{\begin{aligned} 4 x-y+z &=5 \\ 2 y+6 z &=30 \\ x+z &=6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent with no solutions.

Step 1: Arrange the Equations

First, rewrite the system of equations for clarity: \(4x - y + z = 5\), \(2y + 6z = 30\), \(x + z = 6\). We aim to put this system into triangular form, where each equation in sequence contains one more variable than the last.

Step 2: Solve for One Variable in One of the Equations

Choose the third equation \(x + z = 6\) to isolate \(x\). Rearrange it to get \(x = 6 - z\).

Step 3: Substitute to Eliminate a Variable

Substitute \(x = 6 - z\) into the first equation \(4x - y + z = 5\). This results in \(4(6-z) - y + z = 5\) or \(24 - 4z - y + z = 5\). Simplifying gives \(-y - 3z = -19\), which can be rearranged as \(y = -3z + 19\).

Step 4: Substitute into Another Equation

Use \(y = -3z + 19\) in the second equation \(2y + 6z = 30\). Substitute \(y\) to get \(2(-3z + 19) + 6z = 30\), which simplifies to \(-6z + 38 + 6z = 30\). This simplifies further to \(38 = 30\).

Step 5: Analyze Results for Consistency

The equation \(38 = 30\) is false. A false statement indicates that the system of equations is inconsistent, meaning there are no solutions.

Key Concepts

Consistent and Inconsistent Systems

Understanding whether a system of linear equations is consistent or inconsistent is crucial in solving these problems. An inconsistent system is one that does not have any solutions. This happens because the equations represent parallel lines that never intersect, leading to contradictions when solving. In contrast, a consistent system has at least one set of solutions.

• Consistent Independent: A system with exactly one solution. The equations represent lines that intersect at a single point.
• Consistent Dependent: A system with infinitely many solutions. The equations represent the same line, so every point on the line is a solution.
• Inconsistent: A system with no solutions. The equations result in a contradictory statement, such as the false equation we found in the example: \(38 = 30\).

Recognizing these categories helps you understand the nature of the solution. In our exercise, the system was inconsistent because the final step resulted in a false statement.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. These systems are a fundamental part of algebra and appear in numerous mathematical applications.
Each equation in the system corresponds to a line when graphed on a plane. Solving a system of equations means finding all possible points \((x, y, z, \ldots)\) where all the equations hold true simultaneously.
A triangular form in a system of linear equations serves a clear purpose: it simplifies the process of solving the system. This form is characterized by having the equations gradually decrease in complexity as you go down the list. The first equation includes all variables, the second one excludes one, and so on, making it easier to substitute and back-solve.

Substitution Method

The substitution method is a straightforward technique used to find solutions for a system of linear equations. It involves solving one of the equations for one variable, then substituting that expression into the other equations. This process helps reduce the system into simpler equations, making it easier to solve.
In our example, we first isolated \(x\) from the third equation: \(x + z = 6\), which gave us \(x = 6 - z\). This expression for \(x\) was then substituted into the first equation, effectively reducing it to an equation with only two variables, \(y\) and \(z\).
This method allows us to manage complex systems by breaking them down. However, it's crucial to keep track of the equations during substitution to determine consistency and find potential solutions. While this method didn't result in a solution for our exercise due to inconsistency, it still provided valuable insights into the relationship between the variables.