Problem 31
Question
Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{rr} 2 x-3 y-9 z= & -5 \\ x & +3 z=2 \\ -3 x+y-4 z= & -3 \end{array}\right.$$
Step-by-Step Solution
Verified Answer
The system is dependent with solutions: \( x = 2 - 3z, y = 3 - 5z, z = z \).
1Step 1: Identify the System of Equations
We have the following system of linear equations: 1. \( 2x - 3y - 9z = -5 \)2. \( x + 3z = 2 \)3. \( -3x + y - 4z = -3 \). These three equations with three variables form our system.
2Step 2: Simplify the Equation System
Notice in equation 2 we can solve for \( x \):\[ x + 3z = 2 \rightarrow x = 2 - 3z \]Substitute this expression for \( x \) in equations 1 and 3.
3Step 3: Substitute \( x = 2 - 3z \) into Equation 1
Replacing \( x \) in equation 1 gives:\[ 2(2 - 3z) - 3y - 9z = -5 \]Simplifying, we have:\[ 4 - 6z - 3y - 9z = -5 \]Combine like terms:\[ 4 - 15z - 3y = -5 \]Subtract 4 from both sides:\[ -15z - 3y = -9 \]Divide through by -3 to simplify:\[ 5z + y = 3 \] (Equation 4)
4Step 4: Substitute \( x = 2 - 3z \) into Equation 3
Do the substitution in equation 3:\[ -3(2 - 3z) + y - 4z = -3 \]This becomes:\[ -6 + 9z + y - 4z = -3 \]Simplify:\[ 5z + y = 3 \]This result is identical to equation 4 found earlier. This confirms the equations are dependent.
5Step 5: Express System with Parameter
Express \( y \) from the relationship found:\[ y = 3 - 5z \]Thus the solutions can be expressed in terms of the parameter \( z \):1. \( x = 2 - 3z \)2. \( y = 3 - 5z \)3. \( z = z \)This implies the system is dependent and has infinitely many solutions.
Key Concepts
Dependent systemInconsistent systemParameterization
Dependent system
In a system of linear equations, a dependent system occurs when there is at least one equation that can be written as a linear combination of others in the system. This means that not all equations provide new information about the solutions. Essentially, the equations overlap in such a way that they do not independently constrain the values of the variables, leading to an infinite number of solutions.
For example, in the original exercise, equations 1 and 3 were reduced in such a way that the resulting equation (5z + y = 3) was the same, indicating dependency.
To determine dependency, we often look for a scenario where multiple equations simplify into one shared equation. This tells us that the equations are not fully independent, as they describe the same geometric plane in the solution space.
For example, in the original exercise, equations 1 and 3 were reduced in such a way that the resulting equation (5z + y = 3) was the same, indicating dependency.
To determine dependency, we often look for a scenario where multiple equations simplify into one shared equation. This tells us that the equations are not fully independent, as they describe the same geometric plane in the solution space.
- These systems are characteristic of having infinitely many solutions.
- Graphically, this means that the lines or planes represented by the equations overlap completely or partially.
Inconsistent system
An inconsistent system of linear equations is one in which no possible combination of variables satisfies all equations simultaneously. In simpler terms, there's no solution that the system shares.
Inconsistency often arises when trying to solve the equations results in a contradiction, such as a situation where two simplified equations imply 0 = 1 or another false statement.
For example, had our exercise resulted in contradictory statements after substitution, we would deem it inconsistent. Graphically, this means that the lines or planes represented by the equations do not intersect at a single point, expressing inconsistency as parallel lines or planes that do not meet.
When working with systems of equations, recognizing inconsistencies helps problem solvers avoid pursuing solutions that simply do not exist.
Inconsistency often arises when trying to solve the equations results in a contradiction, such as a situation where two simplified equations imply 0 = 1 or another false statement.
For example, had our exercise resulted in contradictory statements after substitution, we would deem it inconsistent. Graphically, this means that the lines or planes represented by the equations do not intersect at a single point, expressing inconsistency as parallel lines or planes that do not meet.
When working with systems of equations, recognizing inconsistencies helps problem solvers avoid pursuing solutions that simply do not exist.
- No solutions exist for inconsistent systems.
- Indicated by contradictions during simplification.
Parameterization
Parameterization is a method used to represent the infinite solutions of a dependent system using one or more free variables, often called parameters. In cases where the system has an infinite number of solutions, such as our dependent system, parameterization becomes a crucial tool.
In the provided exercise, we identified that z is a free variable or parameter. We used z to express other variables, x and y, allowing us to describe the infinite solutions succinctly:
Parameterization helps us understand and interpret systems of equations efficiently, facilitating a deeper comprehension of multiple solutions in algebraic contexts.
In the provided exercise, we identified that z is a free variable or parameter. We used z to express other variables, x and y, allowing us to describe the infinite solutions succinctly:
- The expression for x became: \( x = 2 - 3z \)
- The expression for y became: \( y = 3 - 5z \)
- The parameter z can take any real value.
Parameterization helps us understand and interpret systems of equations efficiently, facilitating a deeper comprehension of multiple solutions in algebraic contexts.
Other exercises in this chapter
Problem 31
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