Problem 10 - Beginning and Intermediate Algebra

Question

Solve the system of equations. $$\begin{aligned} x+3 y+4 z &=1 \\ 3 x+4 y+5 z &=3 \\ x+8 y+11 z &=2 \end{aligned}$$

Step-by-Step Solution

Answer

The given system of linear equations has no solution, as the simplification process led to the false statement \(0=1\). This indicates that the three planes represented by the equations do not have a common intersection point.

Step 1: Write down the given system of equations

We have the following system of linear equations: \(\\ \begin{aligned} x+3 y+4 z &=1 \quad (1)\\ 3 x+4 y+5 z &=3 \quad (2)\\ x+8 y+11 z &=2 \quad (3)\\ \end{aligned} \\ \)

Step 2: Simplify the equations by eliminating one variable

To simplify this system, we can eliminate the variable x by doing the following operations: Subtract equation (1) from equation (3) which gives us a new equation (4): \(\\ \begin{aligned} (3) - (1) \quad &\Rightarrow \quad 5y + 7z = 1 \quad (4) \\ \end{aligned} \\ \) Similarly, subtract equation (1) multiplied by 3 from equation (2) which gives us a new equation (5): \(\\ \begin{aligned} (2) - (1) \cdot 3 \quad &\Rightarrow \quad -5y - 7z = 0 \quad (5) \\ \end{aligned} \\ \)

Step 3: Solve the simplified system

Now we have a simplified system of two linear equations involving two variables y and z: \(\\ \begin{aligned} 5y + 7z &= 1 \quad (4) \\ -5y - 7z &= 0 \quad (5) \\ \end{aligned} \\ \) Now, add equation (4) and equation (5): \(\\ \begin{aligned} (4)+(5) \quad &\Rightarrow \quad 0 = 1 \end{aligned} \\ \)

Step 4: Analyze the outcome

As we obtained 0 = 1, which is a false statement, it means our given system of linear equations has no solution. This can be interpreted as the given equations representing three planes in space that don't intersect at a common point.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that shows the equality of two expressions by using linear functions. These equations can have one, two, or more variables, and they are often represented in the format of straight lines when graphed on a coordinate plane.
For example, the equation \(x + 3y + 4z = 1\) is a linear equation with three variables. Each term in this equation is either a constant or a product of a constant and a variable raised to the power of one.
Linear equations can be combined to form systems of linear equations, which can then be solved to find values of the variables that satisfy all the equations simultaneously.

Solving Systems

A system of equations is a set of two or more equations with the same variables. The objective of solving these systems is to find values for the variables that satisfy all the equations at once.
There are several methods to solve systems of equations, including substitution, graphing, and the elimination method. In our example, the system of equations involves three variables \(x, y, \text{and} z\) and is solved by eliminating variables to simplify the problem.
By going through each step logically, we can reduce the complex system to simpler forms that are easier to solve. Ultimately, we aim to either find a specific solution, find that there are infinitely many solutions, or determine that no solution exists.

No Solution

A system of equations might have no solution, meaning there are no sets of values for the variables that can satisfy all equations simultaneously.
This situation occurs when the equations represent lines or planes that do not intersect. In our solved example, we reached a contradiction, namely \(0 = 1\), indicating that there are no common solutions.
Graphically, this suggests that the planes represented by the equations are parallel or not intersecting in a single point, and thus the system cannot be satisfied. Therefore, understanding that a system can have no solution helps us analyze the relationships between the equations effectively.

Elimination Method

The elimination method is a technique used to solve a system of linear equations. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for others.
In our exercise, we eliminated the variable \(x\) by subtracting equations. This resulted in a new set of simpler equations with just two variables.
Using this method required strategic manipulation of the equations, such as multiplying or adding equations, to make it possible to eliminate a variable efficiently. The goal is to isolate one variable at a time to solve the system step by step. However, here it led us to a conclusion that no solutions are possible.