Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{r}x+y-2 z=0 \\ x-y-4 z=0 \\ y+z=0\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \( x = 3z \), \( y = -z \), and \( z = z \) for any real number \( z \).

Step 1: Analyze the System of Equations

We have a system of three linear equations with three variables:1. \( x + y - 2z = 0 \)2. \( x - y - 4z = 0 \)3. \( y + z = 0 \).Our goal is to solve this system to find the values of \(x\), \(y\), and \(z\).

Step 2: Solve Equation 3 for y in terms of z

Equation 3 is \( y + z = 0 \). Let's solve for \( y \):\[y + z = 0\]\[y = -z\]Now we have \( y \) expressed in terms of \( z \): \( y = -z \).

Step 3: Substitute y in Equation 1 and simplify

Substitute \( y = -z \) into Equation 1:\[x + (-z) - 2z = 0\]\[x - z - 2z = 0\]\[x - 3z = 0\]Now we can express \( x \) in terms of \( z \):\[x = 3z\].

Step 4: Substitute y in Equation 2 and simplify

Substitute \( y = -z \) into Equation 2:\[x - (-z) - 4z = 0\]\[x + z - 4z = 0\]\[x - 3z = 0\].This confirms that the expression for \( x \) is consistent: \( x = 3z \).

Step 5: Express the Solution

With \( x = 3z \) and \( y = -z \), we have the solution expressed in terms of \( z \):- \( x = 3z \)- \( y = -z \)- \( z = z \).We can choose any real value for \( z \), and then \( x \) and \( y \) adjust accordingly. For convenience, if \( z = 1 \), then \( x = 3 \) and \( y = -1 \).

Key Concepts

Systems of Linear Equations

When working with systems of linear equations, we're dealing with a collection of two or more equations. Each of these equations contains the same set of variables. In our example, we have three equations with three variables: \( x \), \( y \), and \( z \). Solving the system involves finding the values of these variables that satisfy all equations simultaneously.

Systems can be solved using different methods. The goal is always to reduce complexity and simplify calculations. Potential solutions can be expressed in terms of one or more variables. In this particular system, we have a special case. We notice that by solving these equations, the solutions for \( x \) and \( y \) are expressed in terms of \( z \). Furthermore, solving systems like this can help in different areas such as physics, economics, and engineering, where relationships between variables are described by equations.

To approach a system of linear equations:

• Identify all equations and variables involved.
• Determine the suitable method to simplify and solve the system.
• Find relationships among variables, which could lead to the solution.

Recognizing these relationships helps eliminate variables and reach a comprehensive solution.

Substitution Method

The substitution method is a powerful technique for solving systems of linear equations. The essence of this method is to express one variable in terms of another and "substitute" this expression into other equations. This reduces the number of variables in that equation, simplifying the process.

In this example, we used the substitution method to solve our system of equations. We start by rearranging Equation 3, solving it for \( y \):

• Original equation: \( y + z = 0 \)
• Rearranged to: \( y = -z \)

Once we have \( y \) expressed in terms of \( z \), we substitute \( y = -z \) into the other equations. This helps to eliminate \( y \) and allows us to solve for \( x \) in terms of \( z \).

• Equation 1: \( x + y - 2z = 0 \) becomes \( x - 3z = 0 \)
• Equation 2: \( x - y - 4z = 0 \) also simplifies to \( x - 3z = 0 \)

The substitution confirms consistent expressions for both \( x \) and \( y \) in terms of \( z \). By reducing the number of variables, substitution as a method provides step-by-step clarity, making it easier to find solutions.

Variables in Terms of Another Variable

Expressing variables in terms of another variable is a useful strategy that simplifies complex systems. By focusing on one variable, you can transform a multi-variable problem into a single-variable one, making the analysis more straightforward.

In our system of equations, \( y \) was expressed as \( y = -z \), and \( x \) as \( x = 3z \). This means both \( x \) and \( y \) depend solely on the value of \( z \).

When all variables are expressed in terms of a single variable, any real number can be assigned to that primary variable. This provides a family of solutions rather than a unique solution.

Here's how you determine relationships:

• Solve one equation for one variable in terms of another.
• Substitute this relationship into other equations to eliminate that variable.
• Continue simplifying until all variables are connected to a primary variable.

Being able to express variables in terms of another variable opens up flexible solution paths, useful in analysis and real-world problem solving.