Problem 29 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{aligned} \frac{1}{2} x-y+\frac{1}{2} z &=-4 \\ x+2 y-3 z &=20 \\ -\frac{1}{2} x+3 y+2 z &=0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{13}{2} \), \( y = \frac{3}{2} \), \( z = -\frac{11}{2} \).

Step 1: Simplify the First Equation

First, let's simplify the first equation by eliminating the fractions. Multiply the entire equation by 2 to get: \( x - 2y + z = -8 \).

Step 2: Rewrite the System of Equations

Now we rewrite the system of equations using the simplified first equation: \( \begin{aligned} x - 2y + z &= -8 \ x + 2y - 3z &= 20 \ -\frac{1}{2}x + 3y + 2z &= 0 \end{aligned} \).

Step 3: Eliminate x from Second and Third Equations

To eliminate \( x \) from the second equation, subtract the first equation from the second: \( 4y - 4z = 28 \). To eliminate \( x \) from the third equation, multiply the first equation by \( \frac{1}{2} \) and add it to the third equation: \( \frac{3}{2}y + \frac{3}{2}z = -4 \).

Step 4: Simplify and Solve for y and z

Simplifying the new equations: \( y - z = 7 \) and \( y + z = -4 \). Adding these two equations gives \( 2y = 3 \), so \( y = \frac{3}{2} \). Subtracting the first simplified equation from the second gives \( 2z = -11 \), so \( z = -\frac{11}{2} \).

Step 5: Solve for x

Substitute \( y = \frac{3}{2} \) and \( z = -\frac{11}{2} \) back into the first simplified equation to solve for \( x \): \( x - 2(\frac{3}{2}) - \frac{11}{2} = -8 \Rightarrow x - 3 - \frac{11}{2} = -8 \Rightarrow x = -\frac{13}{2} \).

Step 6: Verify the Solution

Substitute \( x = -\frac{13}{2} \), \( y = \frac{3}{2} \), and \( z = -\frac{11}{2} \) into the original equations to verify the solution satisfies all equations. This ensures that all the steps are correctly followed and that the solution is valid.

Key Concepts

Solving Systems of Equations

Systems of equations consist of multiple equations that share some or all of the same variables. Solving these equations means finding values for the variables that make all the equations true at the same time. There are various methods to solve systems of linear equations.
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Common methods include:

• Substitution: Solve one equation for one variable, then substitute that expression into the other equations.
• Elimination: Add or subtract equations to eliminate a variable, simplifying the system so that you can solve for one variable at a time.
• Matrix Method: Use matrices and operations like row reduction (also known as Gaussian elimination) to solve the system.

The choice of method often depends on the specific system you're dealing with and personal preference.
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This exercise primarily used elimination to simplify the process.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors and matrices. It provides tools to solve systems of linear equations efficiently. In this context, knowing some linear algebra fundamentals is essential.
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Key concepts include:

• Vectors: Quantities defined by magnitude and direction, fundamental in representing solutions.
• Matrices: Rectangular arrays of numbers that can represent systems of linear equations concisely.
• Operations: Matrix addition, multiplication, and transformations like row reduction.

Understanding how to manipulate matrices and vectors can greatly simplify solving systems. It reduces complex systems into manageable parts. Linear algebra provides both visual and algebraic insights, making it a powerful tool in mathematics.

Equation Simplification

Equation simplification is crucial in solving systems efficiently. Simplifying equations often involves basic algebraic manipulations designed to make complex problems more manageable.
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Here are some common techniques:

• Removing Fractions: Multiply both sides by denominators to eliminate fractions.
• Combining Like Terms: Simplifies expressions by merging similar terms.
• Variable Elimination: Adjust equations to cancel out a variable, reducing system complexity.

The first step in this exercise was to eliminate fractions in the first equation. This simplification paved the way for easier manipulation of the equations during later steps. Effective simplification is an essential skill that saves time and reduces error potential in solving systems of equations.