Problem 30 - Algebra and Trigonometry

Question

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-2 y-3 z &=5 \\ 2 x+y-z &=5 \\ 4 x-3 y-7 z &=5 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Write the System of Equations

The given system of equations is: \( x - 2y - 3z = 5 \) \( 2x + y - z = 5 \) \( 4x - 3y - 7z = 5 \)

Step 2: Choose a Method to Solve

We will use the "Elimination Method" to solve this system of equations. This involves eliminating one variable at a time to reduce the system to two equations and then solve for the remaining variables.

Step 3: Eliminate the Variable x

To eliminate \( x \), first multiply the first equation by 2, and subtract it from the second equation: Equation 1 (multiplied by 2): \( 2x - 4y - 6z = 10 \) Equation 2: \( 2x + y - z = 5 \)Subtract: \( (2x + y - z) - (2x - 4y - 6z) = 5 - 10 \) This yields: \( 5y + 5z = 5 \). Divide throughout by 5: \( y + z = 1 \) (Equation 4).

Step 4: Eliminate Variable x from another Pair

Next, to eliminate \( x \) from the third equation, multiply the first equation by 4, and subtract the result from the third equation:Equation 1 (multiplied by 4): \( 4x - 8y - 12z = 20 \) Equation 3: \( 4x - 3y - 7z = 5 \)Subtract: \( (4x - 3y - 7z) - (4x - 8y - 12z) = 5 - 20 \) This yields: \( 5y + 5z = -15 \). Divide throughout by 5: \( y + z = -3 \) (Equation 5).

Step 5: Analyze Equations 4 and 5

We have two equations: Equation 4: \( y + z = 1 \) Equation 5: \( y + z = -3 \)These equations are inconsistent because \( y + z = 1 \) cannot equal \( y + z = -3 \) simultaneously. Therefore, the system does not have any solutions.

Step 6: Conclude the Inconsistency

Since we have arrived at two contradictory equations, the original system is inconsistent and there are no solutions.

Key Concepts

Elimination Method

The elimination method is a powerful algebraic tool used to solve systems of equations. It involves manipulating the equations to remove, or "eliminate," variables one at a time.
This way, it reduces the complexity of the system. The method works by strategically adding or subtracting whole equations, allowing you to cancel out a specific variable across the equations. Here's a simple breakdown of how to apply the elimination method:

• Identify which variable you want to eliminate first.
• Adjust the coefficients of that variable (using multiplication or division) across two equations so that they will cancel each other out when added or subtracted.
• Add or subtract the equations to eliminate the chosen variable.
• Continue this process with the remaining equations until you find the values of all variables.

This method is particularly useful in solving systems of equations, where an exact solution is required, or in determining whether a system is inconsistent, i.e., it has no solution.

Inconsistent Systems

An inconsistent system of equations is a system where no set of values exists that satisfies all the equations simultaneously. In other words, the equations contradict each other. This can often be determined without finding exact values for variables. In the context of linear equations, an inconsistent system occurs when the equations represent parallel lines that never intersect, meaning they cannot share any common solutions. Mathematically, inconsistency is typically revealed through steps such as these:

• Once equations in a linear system are reduced, if a statement like "0 = 5" appears, this signifies inconsistency.
• If, after elimination, you obtain two different results for the same expression (as seen with the equations "y + z = 1" and "y + z = -3"), it indicates an inconsistency.

Detecting inconsistency early saves time and effort in solving linear systems, ensuring the method used is appropriate.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. Solving a system means finding all variable values that make all the equations true simultaneously. There are different types of systems based on their solutions:

• Consistent Systems: These have at least one solution. If there are multiple solutions, such systems are "dependent," meaning there are infinitely many solutions that typically form a line or plane.
• Inconsistent Systems: When evaluating a system reveals no possible solutions, the system is considered inconsistent, indicating parallel lines or non-intersecting planes in space.

System of equations can arise in numerous real-world contexts, such as defining points on maps, determining break-even points in business, or analyzing currents in electrical circuits.

Variable Elimination

Variable elimination refers to the process of removing, or eliminating, one or more variables from a system of equations. This is a stepping stone towards finding a solution, making the system simpler to solve. The general steps in variable elimination involve:

• Choosing a variable to eliminate, often guided by which variable appears easiest to remove due to its coefficients.
• Manipulating equations to match coefficients for the target variable.
• Adding or subtracting equations to cancel the selected variable out of the equations.
• Repeating the process for other variables as necessary, proceeding to solve the simpler resulting system.

Variable elimination is an efficient algebraic method that aids in solving complex systems, breaking them down into manageable parts. It is an essential skill in algebra, preparing you for more advanced mathematical problem-solving.