Problem 26 - Precalculus: Mathematical for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} x &+3 z=3 \\ 2 x+y-2 z &=5 \\ -y+8 z &=8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is dependent; solutions are parametrized by \( z \).

Step 1: Identify the system of equations

The given system of equations is: \( \text{(1) } x + 3z = 3 \), \( \text{(2) } 2x + y - 2z = 5 \), \( \text{(3) } -y + 8z = 8 \). We need to determine if this system is inconsistent or dependent. If dependent, find the complete solution.

Step 2: Express y from the third equation

From equation (3), \(-y + 8z = 8\), we solve for \( y \): \( y = 8z - 8 \). We will use this expression for substitution into other equations.

Step 3: Substitute y in the second equation

Substitute \( y = 8z - 8 \) into equation (2):\[ 2x + (8z - 8) - 2z = 5 \]. Simplify to get:\[ 2x + 6z - 8 = 5 \]. Further simplification gives:\[ 2x + 6z = 13 \].

Step 4: Express x from simplified second equation

From \( 2x + 6z = 13 \), we solve for \( x \):\[ 2x = 13 - 6z \]\[ x = \frac{13 - 6z}{2} \].

Step 5: Find general solution for dependent system

Using the expression for \( x \) and \( y \) derived from earlier steps, express the solution parametrically in terms of \( z \), which is the free variable:\[ x = \frac{13 - 6z}{2} \],\[ y = 8z - 8 \],\[ z = z \].This indicates a dependent system with solutions determined by the parameter \( z \).

Step 6: Verify consistency across original equations

Verify consistency by substituting the parametrized solutions into the original equations to ensure they hold for \( x = \frac{13 - 6z}{2} \), \( y = 8z - 8 \), \( z = z \). Each original equation is satisfied, confirming the system is dependent.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. The core idea is to find the values of the variables that satisfy all the equations simultaneously. In this exercise, our system of linear equations includes three equations:

• \( x + 3z = 3 \)
• \( 2x + y - 2z = 5 \)
• \( -y + 8z = 8 \)

The objective is to determine the relationship between the equations. In this case, we want to find out if the equations are dependent or inconsistent. Identifying the nature of the system helps us understand whether a unique solution exists or not.

Solutions of Equations

In mathematics, solving a system of equations entails finding values for the variables that make each equation true simultaneously. Let's delve into how we solve the equations given in the exercise. We begin by simplifying equations where possible.

• From equation (3), express one variable in terms of others. Here, solving for \( y \) gives: \( y = 8z - 8 \).
• Substitute back into equation (2) to express another variable. This gives an expression for \( x \), simplifying our system further.

Such techniques help reveal relationships and simplify calculations, easing the path toward finding solutions. The solutions indicate whether a unique, infinite, or no solution exists.

Parametric Solutions

When a system of equations is dependent, the solutions involve parameters, typically because there are fewer independent equations than variables. In our exercise, the variable \( z \) becomes a free parameter. The parametric solution is mapped out as follows:

• \( x = \frac{13 - 6z}{2} \)
• \( y = 8z - 8 \)
• \( z = z \)

These expressions indicate that \( x \), \( y \), and \( z \) are expressed in terms of \( z \). Parametric solutions are useful for representing infinite solutions, where varying the parameter corresponds to different solutions each satisfying all equations in the system.

Consistency in Equations

A consistent system of equations has at least one solution, while an inconsistent system has none. Consistency is verified by substituting the solutions back into the original equations to check for validity.
In the exercise, our approach involved finding solutions in terms of \( z \) and using this information to verify the equations. By substituting \( x = \frac{13 - 6z}{2} \), \( y = 8z - 8 \), and \( z = z \) into each original equation, we observed that all conditions hold true.
This validation shows that the system is consistent and indeed dependent. With each equation satisfied, we conclude that the system's solutions are infinite, generated through the parameter \( z \). Thus, understanding consistency is crucial for categorizing systems and verifying solutions.