Problem 35 - Precalculus with Limits

Question

Solving a System of Linear Equations In Exercises \(25 - 46\) , solve the system of linear equations and check any solutions algebraically. $$ \left\\{ \begin{array} { r r } { x + 4 z = } & { 1 } \\ { x + y + 10 z = } & { 10 } \\ { 2 x - y + 2 z = } & { - 5 } \end{array} \right. $$

Step-by-Step Solution

Answer

The system of equations does not have any solutions as it reaches a contradiction in calculation, and hence it is inconsistent.

Step 1: Simplify the Equations

This set of equations can help. The first equation: \(x + 4z = 1\), the second: \(x + y + 10z = 10\), and third: \(2x - y + 2z = -5\). However, the equations can be simplified. For example, the second equation can be rewritten by subtracting the first equation from it. This gives: \(y + 6z = 9\). The third equation simplified by subtracting the double of the first equation would result in: \(-y - 6z = -7\).

Step 2: Solve the Simplified System

The simplified system of equations is now: \(x + 4z = 1\), \(y + 6z = 9\) and \(-y - 6z = -7\). By adding the second and third simplified equations, we cancel out y and get \(0 = 2\), which is a contradiction.

Step 3: Interpret the Result

The contradiction \(0 = 2\) indicates that the system of equations does not have any solutions. This means that the system of equations is inconsistent, and it does not cross at any point in three-dimensional space which represents that there are no shared solutions among these three equations.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The idea is to find a common solution that satisfies all equations within the system, typically represented as points of intersection when graphed.

Solving such a system gives us the values of the variables that work for all the equations simultaneously. In the case of two variables, this solution can be visualized as the point where two lines cross on a graph. With three variables, we're looking for the point where three planes intersect. However, not all systems have a neat solution: some may have no solution, while others have infinitely many solutions, depending on how the equations relate to each other.

Algebraic Methods

To solve systems of linear equations, we can use algebraic methods such as substitution, elimination, and matrix operations. Substitution involves solving one equation for a variable and then 'substituting' this into another equation. Elimination involves adding or subtracting equations to eliminate a variable and find the value of others. Matrix operations, such as Gaussian elimination, rely on transforming the system into an upper triangular matrix to solve for the variables.

The choice of method often depends on the particular system's characteristics. For example, if one equation is easily solved for one variable, substitution might be the quickest method. On the other hand, if the coefficients of one variable are the same or opposites, elimination could be the most straightforward path.

Inconsistent Systems

Inconsistent systems are systems of equations that do not have a solution. This occurs when the equations represent lines or planes that never intersect, such as parallel lines that have equal slopes. Inconsistencies arise when the equations contradict each other, as seen when manipulating equations to attempt a solution.

For example, if we simplify a system of equations and obtain a false statement such as \(0 = 2\), this is a clear indication of inconsistency. No values for the variables can make all the original equations true at the same time. In the context of an application, this might mean that the set conditions described by the equations cannot be met simultaneously.

Graphical Representation of Linear Equations

The graphical representation of linear equations involves plotting equations on a graph to visualize their solutions. The graph of a linear equation in two variables is a straight line, and in three variables is a plane. The points where these lines or planes intersect correspond to the solutions of the system.

Graphing provides a powerful visual insight into the relationship between equations in a system. If lines intersect at a single point, the system has a unique solution. If they coincide, they have infinitely many solutions. But if they are parallel and do not intersect, as with inconsistent systems, there is no solution. Understanding these patterns can be helpful in analyzing and predicting the nature of the solutions without fully solving the system algebraically.