Problem 13 - Precalculus: Mathematical for Calculus

Question

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$\left\\{\begin{aligned} 2 x-y+3 z &=& 2 & \\ x+2 y-z &=& 4 & & \text { Eliminate the } x \text { -term } \\ -4 x+5 y+z &=& 10 & & \text { from the third equation. } \end{aligned}\right.$$

Step-by-Step Solution

Answer

In the new system, \(x\) is eliminated from the third equation: \(x + 2y - z = 4\) and \(3y + 7z = 14\).

Step 1: Identify Elimination Target

We aim to eliminate the \(x\)-term from the third equation, \(-4x + 5y + z = 10\).

Step 2: Determine Multipliers for Elimination

To eliminate the \(x\)-terms, compare the coefficients of \(x\) in the first equation (\(2x\)) and the third equation (\(-4x\)). Multiply the first equation by 2 to get \(4x\). The current third equation already has \(-4x\). Adding these equations will eliminate \(x\).

Step 3: Multiply First Equation

Multiply the first equation by 2: \(2(2x - y + 3z) = 2(2)\). This results in the new equation: \(4x - 2y + 6z = 4\).

Step 4: Add Equations to Eliminate \(x\)

Add the modified first equation (\(4x - 2y + 6z = 4\)) to the third equation (\(-4x + 5y + z = 10\)). This results in: \(0x + 3y + 7z = 14\).

Step 5: Write New Equivalent System

Combine the unchanged second equation with the newly found equation: \[\begin{aligned} \text{1. } & x + 2y - z = 4 \ \text{2. } & 3y + 7z = 14 \end{aligned}\]

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It involves adding or subtracting equations to remove one of the variables, hence simplifying the system. This approach is particularly effective when dealing with linear equations where coefficients can be manipulated to cancel out variables.

To apply the elimination method, follow these basic steps:

• Identify the variable to eliminate. In this exercise, the target is the \( x \)-term in the third equation.
• Determine the appropriate multipliers to align the coefficients for cancellation. Here, the first equation is multiplied by 2 to match the \( -4x \) term from the third equation.
• Add or subtract the equations as determined from the multipliers to eliminate the chosen variable.

By using these steps, the system is simplified and becomes easier to solve. It's important to work methodically and check each step to ensure that the elimination is accurate.

Linear Equations

Linear equations form the foundation of the given system in this problem. They are equations where the highest power of any variable is one, making them 'linear'. Understanding linear equations is key when solving systems as they contain one or more variables with defined linear relationships.

Each equation in a system of linear equations represents a line in a coordinate space. For example, the original set of equations:

• \( 2x - y + 3z = 2 \)
• \( x + 2y - z = 4 \)
• \( -4x + 5y + z = 10 \)

are all linear, which means any solution must satisfy all equations simultaneously. When dealing with systems like these, each equation can visualize as a plane in three-dimensional space. The solution is where all three planes intersect, representing the values of \( x, y, \) and \( z \) that satisfy all.

Algebraic Manipulation

Algebraic manipulation is the process used to simplify equations and solve them efficiently. It involves rearranging terms, applying arithmetic operations, and using properties of equality to maintain equivalence while transforming an equation.

In the provided solution, several algebraic manipulations are used:

• Multiplying the first equation by 2 changes it to \( 4x - 2y + 6z = 4 \). This step ensures that the \( x \) coefficients align for elimination.
• Adding the modified first equation to the third equation to eliminate \( x \): \(-4x + 5y + z = 10\), resulting in \( 3y + 7z = 14 \).

These changes are crucial for transforming the original system into a simpler form without losing the inherent relationships. Algebraic manipulation requires careful handling to preserve the equation's balance, ensuring each transformation leads closer to a solution.