Problem 29 - Algebra and Trigonometry

Question

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} 2 x+3 y-z &=1 \\ x+2 y &=3 \\ x+3 y+z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations is inconsistent.

Step 1: Write the System of Equations

The given system of equations is:1. \( 2x + 3y - z = 1 \)2. \( x + 2y = 3 \)3. \( x + 3y + z = 4 \)

Step 2: Solve for x in the Second Equation

From the second equation, \( x + 2y = 3 \), solve for \( x \):\[x = 3 - 2y\]

Step 3: Substitute x in the First Equation

Substitute \( x = 3 - 2y \) into the first equation:\[2(3 - 2y) + 3y - z = 1 \]Simplify:\[6 - 4y + 3y - z = 1 \]\[6 - y - z = 1 \]Rearrange:\[z = 5 - y\]

Step 4: Substitute x in the Third Equation

Substitute \( x = 3 - 2y \) into the third equation:\[(3 - 2y) + 3y + z = 4\]Simplify:\[3 + y + z = 4\]Rearrange:\[y + z = 1\]

Step 5: Substitute z from Step 3 into Step 4

Substitute \( z = 5 - y \) into the equation \( y + z = 1 \):\[y + (5 - y) = 1\]\[5 = 1\]This is a contradiction, which means the system does not have a consistent solution.

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations with the same variables. For example, if we have three equations with variables \(x\), \(y\), and \(z\), our goal is to find a set of values for these variables that satisfies all the equations simultaneously. Each equation in the system represents a condition that must be met.These systems can be classified based on their solutions:

• Consistent and independent: Only one solution exists.
• Consistent and dependent: An infinite number of solutions are available (equations coincide).
• Inconsistent: No solution satisfies all equations at once.

Being able to determine the type of system is crucial for finding answers effectively.

Inconsistent Systems

An inconsistent system of equations is one where no solution exists that satisfies all equations in the system. This happens when the equations represent parallel lines that never intersect or reach a contradictory result.When solving a system, if you end up with a statement like \(5 = 1\) or any other false statement, it indicates that the system is inconsistent. In our given example, the final step led to this result, showing there are no values for \(x\), \(y\), and \(z\) that can make all the original equations true simultaneously. Recognizing such contradictions is key when solving systems of equations.

Substitution Method

The substitution method is a way to solve systems of equations that involves solving one of the equations for one variable, then substituting that expression into the other equations. This process helps eliminate one variable at a time, making the system easier to solve.Here's how it works:

• Solve one of the equations for one variable.
• Substitute the expression for that variable into the other equations.
• Simplify and solve for the remaining variables.

For our exercise, we first solved the second equation for \(x\), obtaining \(x = 3 - 2y\), then substituted this expression into the first and third equations. This method is particularly effective for systems where at least one equation is simple to manipulate.

Solving Linear Equations

Solving linear equations involves finding the variable values that make the equation true. In a single equation, this typically involves isolating the variable on one side. For multiple linear equations like in a system, it involves methods such as substitution or elimination to reduce the number of variables step by step. In our textbook example, the equations were manipulated using substitution to eventually spot an inconsistency, leading to a conclusion that no solution exists for the system. Key steps include:

• Rearranging terms to isolate variables.
• Simplifying to find relationships between variables.
• Checking the solutions in all original equations to ensure consistency.

Understanding these steps helps ensure that you correctly solve or recognize the nature (such as inconsistency) of the system you're working with.