Problem 37 - Algebra and Trigonometry

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{array}{l}{2 x+y-2 z=12} \\ {-x-\frac{1}{2} y+z=-6} \\ {3 x+\frac{3}{2} y-3 z=18}\end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent; solutions are parameterized by \((x, 12 - 2x + 2t, t)\).

Step 1: Express System in Augmented Matrix Form

Write the given system of equations in an augmented matrix form:\[\begin{bmatrix} 2 & 1 & -2 & | & 12 \ -1 & -\frac{1}{2} & 1 & | & -6 \3 & \frac{3}{2} & -3 & | & 18\end{bmatrix}\]

Step 2: Row Reduce to Row-Echelon Form

Use row operations to transform the matrix to row-echelon form. Start by eliminating the first column below the pivot in the first row:1. Multiply the second row by 2 and add it to the first row: \[ \begin{bmatrix} 2 & 1 & -2 & | & 12 \ 0 & 0 & 0 & | & 0 \ 3 & \frac{3}{2} & -3 & | & 18 \end{bmatrix} \]2. Eliminate the first column in the third row by multiplying the first row by -3/2 and adding it to the third row: \[ \begin{bmatrix} 2 & 1 & -2 & | & 12 \ 0 & 0 & 0 & | & 0 \ 0 & 0 & 0 & | & 0 \end{bmatrix} \]

Step 3: Interpret Row-Echelon Form

Analyze the resulting row-echelon form. The third row is all zeros, indicating redundancy. The second row is also all zeros, confirming there are infinitely many solutions. Thus, the system is dependent.

Step 4: Solve for Dependent Variables

Use the non-zero row to express one variable in terms of the others. From the first row \(2x + y - 2z = 12\), express: \[ y = 12 - 2x + 2z \]Thus the variables can be expressed in terms of a parameter \(z=t\):- Let \(z = t\).- Then \(x\) is any real number subject to \(2x + y - 2t = 12\).- \(y = 12 - 2x + 2t\).Thus, any solution can be written as:\[(x, y, z) = (x, 12 - 2x + 2t, t), \text{ where } x, t \in \mathbb{R}.\]

Step 5: Conclusion

The system is dependent with solutions as parameterized linear combinations of the variable \(z\).

Key Concepts

Dependent System

In linear algebra, a system of linear equations can fall into different categories: consistent, inconsistent, or dependent. A dependent system is a set of equations where all the equations are essentially the same line in a geometric sense. This means that each equation in the system provides the same solution or can be derived from one another. This type of system doesn't have a unique solution but rather infinitely many solutions, which satisfy all the given equations in the system.
For example, if one equation in the system is a multiple or a linear combination of another, the system is dependent. This occurs because the lines represented by the equations overlap entirely. Therefore, instead of intersecting at a single point, they coincide, leading to infinitely many solutions that lie along the line they form.

Augmented Matrix

An augmented matrix is a compact and organized way to represent a system of linear equations. By turning a system of equations into an augmented matrix, we can use matrix operations to solve it more efficiently. An augmented matrix is formatted to combine the coefficients of the variables and the constants from the equations.
For instance, given the linear equations:

• 2x + y - 2z = 12
• -x - \(\frac{1}{2}\)y + z = -6
• 3x + \(\frac{3}{2}\)y - 3z = 18

This system can be expressed in the augmented matrix form as:\[\begin{bmatrix}2 & 1 & -2 & | & 12 \-1 & -\frac{1}{2} & 1 & | & -6 \3 & \frac{3}{2} & -3 & | & 18\end{bmatrix}\]The vertical bar separates the coefficients of the variables from the constants. This matrix form makes it simpler to perform row operations to solve or simplify the system.

Row-Echelon Form

The row-echelon form is a specific arrangement of a matrix used to simplify solving linear equations. It's achieved through a series of elementary row operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiples of rows from others.
In the row-echelon form:

• All rows having leading entries (the first non-zero number from the left in a row) are arranged such that they progress diagonally downwards to the right.
• Any zero rows are placed at the bottom of the matrix.
• The leading entry of a non-zero row is always to the right of the leading entry in the row above.

For the system being analyzed, transforming it to row-echelon form involved eliminating variables using row operations, resulting in a matrix where lower rows are zero, suggesting dependent relations among the equations, thus indicating infinitely many solutions.

Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, it means there are infinitely many sets of values for the variables that satisfy all the equations. This scenario typically arises in dependent systems, where the equations define overlapping lines in the solution space.
Infinitely many solutions are often found by expressing the variables in terms of free parameters (like a parameter \(t\)). In our example, we concluded that for one of the variables expressed as \(z = t\), others can be written in terms of it:

• \(x\) can be any real number, constrained by equations, \(2x + y - 2t = 12\)
• \(y = 12 - 2x + 2t\)

Hence, the solutions can be parameterized, and any point \((x, y, z)\) satisfying \(x\) and parameter \(t\) leads to a valid solution. This representation shows how solutions form a line or a plane in multidimensional space, indicative of dependency within the system.