Problem 40 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{rr} x-y-2 z= & -11 \\ x-2 y-z= & -11 \\ -x+y+3 z= & 14 \end{array} $$

Step-by-Step Solution

Answer

\(x = -2\), \(y = 3\), \(z = 3\).

Step 1: Identify the System of Equations

We are given a system of three equations: 1. \( x - y - 2z = -11 \)2. \( x - 2y - z = -11 \)3. \( -x + y + 3z = 14 \)

Step 2: Simplify the System

Let's begin by eliminating one of the variables in order to create a simpler system. We will add equations (1) and (3) to eliminate \( x \):1. \( x - y - 2z = -11 \)3. \( -x + y + 3z = 14 \)Adding these gives: \( (x - y - 2z) + (-x + y + 3z) = -11 + 14 \), which simplifies to \( z = 3 \).

Step 3: Substitute and Simplify

Now that we know \( z = 3 \), substitute \( z \) into one of the original equations to find another variable. We'll use equation (1):\( x - y - 2(3) = -11 \)This simplifies to: \( x - y - 6 = -11 \)Rearranging gives: \( x - y = -5 \). (Let's refer to this as equation 4.)

Step 4: Substitute in another Equation

Now, use \( z = 3 \) in equation (2) to find a relationship between \( x \) and \( y \):\( x - 2y - 3 = -11 \)This becomes:\( x - 2y = -8 \). (Let's refer to this as equation 5.)

Step 5: Solve the Simplified System

We now have two equations:Equation 4: \( x - y = -5 \)Equation 5: \( x - 2y = -8 \)Subtract equation 4 from equation 5:\( (x - 2y) - (x - y) = -8 - (-5) \)\( -2y + y = -3 \)\( -y = -3 \) Thus, \( y = 3 \).

Step 6: Find the Remaining Variable

Now substitute \( y = 3 \) back into equation 4:\( x - 3 = -5 \)Thus \( x = -5 + 3 \)\( x = -2 \). All variables are now determined.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that denote a straight line when graphed on a coordinate plane. Each equation typically involves variables raised only to the power of one and can be written in the standard form:

• \( ax + by + cz = d \), where \( a, b, \) and \( c \) are constants, and \( d \) is the constant term.

In a system of linear equations, we have multiple such equations specified to solve for shared variables. The goal is to find a set of variable values satisfying all equations simultaneously.
The given system involves three equations with three variables \( x, y, \) and \( z \), making it a classic problem of three-dimensional linear systems. Each equation provides geometrical information about the plane in space, and solving the system determines the intersection, representing the solution.

Variable Elimination

Variable elimination is a powerful technique used in solving systems of equations. The main idea is to eliminate one variable by combining two equations so that the resulting system becomes simpler. Here's how it works:

• Choose a variable to eliminate. In this exercise, \( x \) was chosen.
• Combine equations that will cancel out this variable. By adding the first and third equations, \( x \) is successfully eliminated.

After eliminating a variable, you reduce the number of variables and equations in the system. This leads to simpler calculations.
For example, adding equations (1) and (3): \( (x - y - 2z) + (-x + y + 3z) = -11 + 14 \)
Produces \( z = 3 \), simplifying our system significantly.

Substitution Method

After using variable elimination, the substitution method serves as the next step to solve for unknowns. Once you find the value for one variable, like \( z = 3 \), substitution can be used to find others:

• Substitute the value of one variable into the remaining equations. This process gradually unveils the values of other variables.

For instance, replacing \( z = 3 \) in the first equation, we derive \( x - y = -5 \). Substitution reduces the complexity of systems, turning a potentially difficult problem into manageable pieces. The key is to carefully replace values to make further calculations straightforward and methodical.

Algebraic Simplification

Algebraic simplification is a fundamental process in solving equations. It involves rewriting expressions in simpler forms to make them easier to solve. During this task:

• Combine like terms and constants for a straightforward equation.
• Rearrange equations to isolate variables.

In this example, after finding \( y = 3 \) through simplification, substituting it back into the derived equation \( x - y = -5 \) gives us \( x = -2 \). Simplification helps finalize the solution by clarifying the relationships between variables.
It is important to maintain neat and expanded work processes to prevent errors and maintain clear visibility of each variable's role in achieving the solution. In summary, simplify wherever possible to streamline solving steps effectively and efficiently.