Problem 35 - Elementary and Intermediate Algebra

Question

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} x \quad \quad+z &=0 ,\\\ x+y+2 z &=3 ,\\\ y+ \quad z &=2 \end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Write down the system of equations

The system of equations is given as: \[ \begin{aligned} x + z &= 0, \ x + y + 2z &= 3, \ y + z &= 2. \end{aligned} \]

Step 2: Isolate one variable from the first equation

From the first equation, solve for \(x\): \[ x + z = 0 \rightarrow x = -z. \]

Step 3: Substitute the isolated variable into the second equation

Substitute \(x = -z\) into the second equation: \[ (-z) + y + 2z = 3 \rightarrow y + z = 3. \]

Step 4: Compare with the third equation

Notice that the resulting equation from step 3 is the same as the third equation: \[ y + z = 3 eq y + z = 2 \]which means the system of equations is inconsistent and there is no solution.

Key Concepts

Solving Equations

Solving equations involves finding the values of the variables that make the equations true. For a system of equations, we look for values that satisfy all equations simultaneously. This means we need a set of values for variables like x, y, and z that work in every equation given.

To solve, we follow these steps:

• Write down all the equations in the system.
• Isolate one variable from one equation.
• Substitute that isolated variable into the other equations.
• Simplify and solve the resulting equations.
• Check your solutions by plugging them back into the original equations.

Simplifying and substituting helps to reduce the system to fewer equations with fewer variables, making it more manageable. This process is useful whether solving simple or complex systems.

Inconsistent Systems

An inconsistent system of equations is one in which no set of variable values can satisfy all the equations simultaneously. In our exercise, we encountered this exact situation. Here’s how it was identified:

• We simplified the first equation to find x in terms of z.
• After substituting x = -z into the second equation, we derived a new equation y + z = 3.
• Comparing this with the third equation y + z = 2 showed a contradiction.

This contradiction reveals that no values of x, y, and z can satisfy both y + z = 2 and y + z = 3 simultaneously. Therefore, the system is inconsistent, indicating no solution exists.

Inconsistent systems often arise in real-world scenarios where conflicting constraints cannot be simultaneously satisfied, like conflicting schedules or incompatible requirements.

Algebra

Algebra provides the tools to manipulate and solve equations. It involves using variables, constants, and arithmetic operations. When dealing with systems of equations, algebra allows us to rearrange, simplify, and manipulate equations to find solutions.

Key algebraic skills used in solving systems include:

• Substitution: Replacing a variable with an equivalent expression from another equation.
• Elimination: Adding or subtracting equations to remove a variable.
• Simplification: Combining like terms and reducing equations to their simplest forms.

In our system, we utilized substitution and simplification to reveal a contradiction, thereby identifying the system as inconsistent. Mastery of these algebraic techniques is essential for efficiently solving systems of equations and understanding their properties.