Problem 21 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{rr} -2 x-4 y+6 z= & -8 \\ x+2 y-3 z= & 4 \\ 4 x+8 y-12 z= & 16 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \( (x, y, z) = (4 - 2t + 3s, t, s) \).

Step 1: Identify Dependencies

Observe the given system of equations:\[ \begin{align*} \text{1.} & -2x - 4y + 6z = -8 \ \text{2.} & x + 2y - 3z = 4 \ \text{3.} & 4x + 8y - 12z = 16 \end{align*} \] Notice that Equation 3 is simply twice Equation 2, which means they are linearly dependent. This reduces our system to two independent equations.

Step 2: Solve for One Variable

From Equation 2: \( x + 2y - 3z = 4 \), we can solve for \( x \):\[ x = 4 - 2y + 3z \] Substitute this expression for \( x \) into Equation 1.

Step 3: Substitute and Simplify

Substitute \( x = 4 - 2y + 3z \) into Equation 1: \[ -2(4 - 2y + 3z) - 4y + 6z = -8 \] Simplify this equation: \[ -8 + 4y - 6z - 4y + 6z = -8 \] Notice the terms cancel each other out: \[ -8 = -8 \] This confirms the equations are consistent and represent the same line.

Step 4: Interpret the Solution

The solution implies that the equations are dependent (describe the same plane), therefore they have infinitely many solutions based on free parameters for \( y \) and \( z \). Pick \( y = t \) and \( z = s \) to represent the free variables and express \( x \) in terms of \( t \) and \( s \): \[ x = 4 - 2t + 3s \]

Step 5: Formulate the Parametric Solution

Write the general solution in parametric form as:\[ \left( x, y, z \right) = \left( 4 - 2t + 3s, t, s \right) \] where \( t \) and \( s \) are parameters representing any real numbers.

Key Concepts

Linear Dependence

In a system of linear equations, some equations might be multiples of others. This means they are linearly dependent. Linear dependence occurs when one equation can be derived from another by multiplying by a constant.
It can reduce the system's complexity, as it essentially provides the same information repeatedly. In the provided exercise, you can see linear dependence between Equation 3 and Equation 2, since Equation 3 is just twice Equation 2:

• Equation 2: \( x + 2y - 3z = 4 \)
• Equation 3: \( 4x + 8y - 12z = 16 \)

This dependence allows us to ignore the redundant Equation 3, simplifying our analysis. Hence, the system is reduced to the independent equations given by Equation 1 and Equation 2. Understanding linear dependence is crucial because it lays the groundwork for identifying the nature of solutions of a system.

Parametric Solution

Once linear dependence reduces our system, we need to solve it. Often, systems with more variables than independent equations require a parametric solution. Parametric solutions express variables in terms of any arbitrary parameters.
In our exercise, with two equations and three variables \((x, y, z)\), we can represent two variables as parameters. By solving \(x + 2y - 3z = 4\) for \(x\), we get:

• \(x = 4 - 2y + 3z\)

Here, we can let \(y = t\) and \(z = s\), where \(t\) and \(s\) are parameters. The existence of parameters shows that the system remains consistent and can describe multiple solutions.The parametric solution is expressed as:

• \((x, y, z) = (4 - 2t + 3s, t, s)\)

Parameters \(t\) and \(s\) represent any real numbers providing different solutions within the same system.

Infinitely Many Solutions

When solving linear systems, you might encounter a consistent solution that isn't unique but rather infinite. This indicates that the system has designed constraints, setting a scenario where infinitely many solutions exist. This occurs when relationships among equations describe a common geometric feature, generally as lines or planes in space.
In the given problem, after identifying linear dependence and using parametric solutions, the system manifests as infinitely many solutions. By expressing \(y = t\) and \(z = s\), with arbitrary \(t\) and \(s\), the problem shows that for every \((t, s)\) pair, an \((x, y, z)\) solution satisfies the system:

• \((x, y, z) = (4 - 2t + 3s, t, s)\)

This denotes a line of solutions in the 3D coordinate system, allowing every pair of \(t\) and \(s\) to create a valid solution. Recognizing where and why infinitely many solutions occur helps in interpreting the nature of the mathematical model the system represents.