Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{r}2 x-y+z=0 \\ x-y-2 z=0 \\ 2 x-3 y-z=0\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \( x = t \), \( y = -t \), \( z = t \) for any real number \( t \).

Step 1: Write Down the System of Equations

The given system of equations is: \( \begin{cases} 2x - y + z = 0 \ x - y - 2z = 0 \ 2x - 3y - z = 0 \end{cases} \). We need to solve for the variables \( x \), \( y \), and \( z \).

Step 2: Eliminate a Variable

Let's eliminate \( z \) by combining the equations. Add the first and second equations: \((2x - y + z) + (x - y - 2z) = 0 + 0\). This results in \(3x - 2y - z = 0\).

Step 3: Solve for One Variable

Now, subtract the third equation from this result: \((3x - 2y - z) - (2x - 3y - z) = 0 - 0\). This simplifies to \(x + y = 0\), or \(x = -y\).

Step 4: Substitute to Find Remaining Variables

Substitute \( x = -y \) into the original equations. Replace \( x \) with \( -y \) in the second equation: \(-y - y - 2z = 0 \Rightarrow -2y - 2z = 0 \Rightarrow y + z = 0\). Thus, \( y = -z \).

Step 5: Express All Variables in Terms of One

Replacing \( y = -z \) and \( x = -y = z \), we represent all variables in terms of \( z \): \( x = z \), \( y = -z \), and use \( z = t \) where \( t \) is any real number.

Step 6: Check the Solution

To verify, substitute \( x = t \), \( y = -t \), \( z = t \) back into the original equations. All three equations simplify to 0, confirming the solution.

Key Concepts

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on vectors, vector spaces, linear mappings, and the systems of linear equations.
For solving a system of equations involving multiple variables, linear algebra simplifies the process by using methods such as Gaussian elimination or matrix operations.
Understanding linear algebra is crucial as it is widely applied in fields such as computer science, engineering, physics, and economics.

• A system of equations like the one given here can be approached using these methods to find a solution that satisfies all equations simultaneously.
• Linear equations are equations of the first degree, meaning they include variables raised only to the first power.

Linear algebra provides tools to manipulate these equations to isolate and solve for the variables involved.

Variables

Variables are symbols or letters like \( x, y,\) and \(z\) used to represent unknown values in an equation or mathematical function.
They are what we solve for in the system of equations.
In our problem, we needed to find values for \( x, y, \) and \( z \) that simultaneously satisfy all three given equations.

• The approach often involves manipulating the equations to express as many variables as possible in terms of one of them, such as finding \( x = -y \).
• In our solution, \( t \) is introduced as a free variable representing \( z \), allowing \( x, y, \) and \( z \) to be expressed in terms of \( t \).

This concept of variables is foundational in algebra, making complex problems more manageable.

Solution Verification

Solution Verification is the process to ensure that the values obtained for variables satisfy the original system of equations.
Verifying solutions helps confirm the accuracy of calculations and methodologies used.
In our exercise, after finding solutions \( x = t, y = -t, z = t \), we substituted these back into the original equations.

• The aim was to see if they simplify to a true statement; in this case, all equations simplified to 0.
• Verification ensures that no steps were overlooked or errors made during calculations, solidifying confidence in the results.

This important step confirms the relationships and dependencies among all variables are correctly defined.

Substitution Method

The Substitution Method involves solving one equation for one variable and then substituting this value into the other equations.
It helps simplify systems of equations, especially when dealing with multiple variables.
In this problem, we found that \( x = -y \) from one of the equations.

• By substituting \(-y\) for \(x\) in the subsequent equations, we progressively simplified the problem, expressing all variables in terms of a single parameter, \( t \).
• This allowed for an elegant, comprehensive solution to the system, showing the power of substitution when coupled with logical problem-solving.

The substitution method is a valuable tool in algebra, offering a systematic way to tackle problems that seem complex at first glance.