Question

Solve the system of linear equations using the substitution method. \(x+y+z=4\) \(5 x+5 y+5 z=12\) \(x-4 y+z=9\)

Short answer

The original system of equations has no solution.

Step-by-step solution

Simplify the equations

First off, simplify the equations if possible. The second equation can be simplified by dividing each term by 5: \(x+y+z=12/5\) which simplifies to \(x+y+z=2.4\)

Apply the substitution method

Now there are two equations with the same variables left, \(x+y+z=4\) and \(x+y+z=2.4\). Subtract the second equation from the first to eliminate \(x\), \(y\), and \(z\) to get \(0=1.6\). This equation is not valid because 0 does not equal 1.6.

Conclusion

This means that the original system of equations has no solution. Thus, the set of equations is inconsistent, and it does not have a solution.

Key concepts

Substitution Method

The substitution method is a key technique utilized in linear algebra to find the values of variables in systems of linear equations. To apply this method, follow the given steps:

1. Choose one of the equations and solve it for one variable in terms of the others.
2. Substitute this expression into the other equations.
3. Continue to solve the resulting equations to eliminate another variable.
4. Repeat the process until all variables are solved for or until you can reach a conclusion about the solution set of the system.

In the context of our given exercise, an attempt was made to apply the substitution method by simplifying one equation and then combining it with another. However, during the process, it was discovered that the resulting equation, after simplification, did not make logical sense, indicating that there was no solution. This crucial point emphasizes the importance of closely examining the results at each stage of substitution to accurately interpret the system's solutions.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. In the practical sense, linear algebra is about solving systems of linear equations and understanding vector spaces and matrices.

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to determine if there's a common solution to all equations, which corresponds to the intersection point(s) of the respective lines, planes, or hyperplanes represented by each equation. Methods such as substitution, elimination, and matrix operations are commonly employed to find these solutions. Through these methods, we can categorize the system of equations as having a unique solution, infinite solutions, or no solution at all.

Inconsistent System

An inconsistent system of equations is one in which no solution satisfies all the equations simultaneously. This occurs when the lines, planes, or hyperplanes that represent the equations do not intersect at any point. For a system of linear equations, there are three broad categories:

• Consistent and independent systems: These have a single unique solution.
• Consistent and dependent systems: These have an infinite number of solutions.
• Inconsistent systems: These do not have any solution at all.

In the exercise provided, the result of the substitution method led to a false statement (\(0=1.6\)), indicating that this system is inconsistent. Therefore, there is no set of values for the variables that will satisfy all three equations. Understanding that an inconsistent system has no solution is vital for students as it helps avoid unnecessary attempts at finding solutions where none exist.