Problem 27 - Algebra and Trigonometry

Question

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y-z &=0 \\ x+2 y-3 z &=-3 \\ 2 x+3 y-4 z &=-3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 3 - z, y = 2z - 3, z = z \), for any real number \( z \).

Step 1: Write Down the System of Equations

The system of equations given is:1. \( x + y - z = 0 \)2. \( x + 2y - 3z = -3 \)3. \( 2x + 3y - 4z = -3 \)

Step 2: Use the Elimination Method

First, eliminate \(x\) from the second and third equations:- Subtract equation 1 from equation 2: \[ (x + 2y - 3z) - (x + y - z) = -3 - 0 \] Simplifying, we get: \[ y - 2z = -3 \] - Subtract equation 1 from equation 3: \[ (2x + 3y - 4z) - (2)(x + y - z) = -3 - 0 \] Simplifying, we have: \[ y - 2z = -3 \]

Step 3: Check for Consistency

We obtained the same equation \( y - 2z = -3 \) from both eliminations, which is consistent. Now substitute \( y = 2z - 3 \) back into one of the original equations.

Step 4: Solve for x, y, z

Substitute \( y = 2z - 3 \) into equation 1:\[x + (2z - 3) - z = 0\]This simplifies to:\[x + z - 3 = 0\]Solving for \(x\), we have:\[x = 3 - z\]Hence, the solution is \( x = 3 - z, y = 2z - 3, z = z \), where \( z \) is a parameter.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In our exercise, we deal with three linear equations sharing the variables \(x\), \(y\), and \(z\). The goal is to find values for these variables that satisfy all equations simultaneously.
Understanding systems of equations involves realizing why we're dealing with more than one equation. Essentially, each equation provides a constraint, or rule, that the solution has to satisfy. Solving the system means finding numbers for \(x\), \(y\), and \(z\) that make all of these equations true at the same time.

• Equations: three separate expressions relating \(x\), \(y\), and \(z\)
• Objective: find values satisfying all equations

For example, "solving" the system here involves identifying the set of variable values that satisfy each of our three original linear constraints. This interaction of constraints is why we work with systems instead of individual equations.

Elimination Method

The elimination method is a technique used to find the solution of a system of equations. By manipulating the equations, we eliminate one variable at a time, making it easier to solve the system.
In the exercise, elimination is used to remove the variable \(x\) from equations. This simplifies the problem to two equations in terms of \(y\) and \(z\):

• Subtracting equation 1 from equation 2 gives: \(y - 2z = -3\).
• Similarly, subtracting a scaled version of equation 1 from equation 3 also leads: \(y - 2z = -3\).

The objective is always to simplify the system using algebraic operations, ultimately leveraging simplicity to reach the solution more straightforwardly.

Consistent System

A system of equations is referred to as consistent if there is at least one solution that satisfies all the equations. Conversely, it is inconsistent if no single solution can satisfy all equations.
In our particular exercise, the system is consistent. By employing the elimination method, we obtain the same equation repeated, \(y - 2z = -3\), twice. This repetition indicates consistency, as it shows alignment between the equations. When all rewritten versions of the original equations align like this, it confirms that we're on track to fully solving the system.

• Consistent systems have one or more solutions.
• Inconsistent systems have no solutions, typically arising from contradictory equations.

Verifying consistency is thus a crucial step when working with systems of equations.

Parametric Solution

A parametric solution expresses the solution of a system of equations using parameters, often introducing one or more free variables. This is especially useful for linear systems where infinite solutions may occur.
In our exercise, after substitution and elimination, we're left with expressions for \(x\) and \(y\) in terms of \(z\):

• \(x = 3 - z\)
• \(y = 2z - 3\)
• \(z = z\)

Here, \(z\) acts as a parameter, meaning it can take any real number value. This leads to a family of solutions rather than a single solution.
When systems have infinite solutions, it's common to utilize parametric forms, capturing the range of all possible solutions in a simple, structured format.