Problem 5 - College Algebra

Question

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} x+y+2 z=11 \\ x+y+3 z=14 \\ x+2 y-z=5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\), \(y = 3\), and \(z = 3\).

Step 1: Organize the Equations

First, write down the three given equations in a clear, organized way:\n\[\begin{array}{ccc} x & +y & +2z = 11 \ x & +y & +3z = 14 \ x & +2y & -z = 5 \end{array}\]

Step 2: Perform Gaussian Elimination

To simplify the system of equations into an upper triangular matrix, we subtract the first equation from the second and second from the third. This gives us:\n\[\begin{array}{ccc} x & +y & +2z = 11 \ 0 & 0 & z = 3 \ 0 & y & -3z = -6 \end{array}\]

Step 3: Solve for the Variables

With the simplified system of equations, we can now solve for our variables. We can clearly see that \(z = 3\) from the second equation. Substitute \(z = 3\) into the first and third equations, we can solve for \(x\) and \(y\):\n\[x + y = 5\]\n\[x + 2y = 8\]\nSubtract the former from the latter, we get \(y = 3\). Substitute \(y = 3\) into the equation \(x + y = 5\), we can solve for \(x\), which gives \(x = 2\).

Key Concepts

Gaussian Elimination

Gaussian Elimination is a powerful algebraic method used to solve systems of linear equations. It involves performing operations on the rows of a matrix representing the system, with the goal of transforming it into an upper triangular form. This means that below the main diagonal of the matrix, all the elements are zeros.

To illustrate this with an example, let's look at the step-by-step solution of our provided exercise. The system initially consists of three equations with three variables (x, y, z). Through Gaussian Elimination, the second and third equations are manipulated by subtracting the other equations, which eliminates certain variables and simplifies the system.

Once the system is simplified, it's easier to solve since we can start by finding the value of z from an equation where it is the only variable left, and then use back-substitution to solve for y and x. It is critical in this process to organize the equations neatly and ensure that every step is done correctly to avoid errors that can compromise the solution.

System of Linear Equations

A system of linear equations consists of two or more linear equations that share a set of variables and are considered simultaneously. Solving a system means finding values for the variables that satisfy all equations in the system at once.

The system provided in the exercise contained three equations, each with x, y, and z variables. When graphed, each equation represents a plane in three-dimensional space, and the solution to the system is the point (or points) where these planes intersect. Systems of linear equations can have one unique solution, no solution if the planes are parallel and do not intersect, or infinitely many solutions if the planes coincide.

The step-by-step solution showed the process of finding a unique solution, confirming that the planes intersect at exactly one point in space. Applying a methodical approach to these problems is essential, as it helps in visualizing and understanding the geometric interpretation of these equations.

Algebraic Methods

Algebraic methods, such as substitution, elimination, and matrix operations like Gaussian Elimination, are key techniques for solving systems of linear equations. These methods convert the system into forms that are easier to manipulate, thus facilitating the determination of the variables' values.

In the context of the given exercise, Gaussian Elimination was the chosen algebraic method, but others could potentially be used. Substitution involves expressing one variable in terms of the others and then substituting that expression into the remaining equations. Elimination, similar to Gaussian Elimination, focuses on removing variables from equations to reduce the complexity.

No matter which algebraic method is used, it's vital to maintain the balance of the equations by performing the same operations on both sides. This balance ensures the integrity of the solution and is a fundamental principle in algebra. Choosing the most efficient method can vary depending on the nature of the system and the preference of the person solving it.