Problem 15 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The new system is: \(2x - y + 3z = 2\), \(x + 2y - z = 4\), \(13y - 3z = 26\).

Step 1: Choose an Operation to Eliminate the x-Term

To eliminate the \(x\)-term in the third equation \(-4x + 5y + z = 10\), we first need to decide on a suitable operation. We can either multiply the first or second equation to have a \(4x\) or \(-4x\) respectively, and then add it to the third equation to cancel out the \(x\)-term.

Step 2: Multiply and Prepare for Elimination

Choose the second equation \(x + 2y - z = 4\) to manipulate. We multiply it by 4 to match the \(-4x\) of the third equation:\[4(x + 2y - z) = 4(4)\]This simplifies to:\[4x + 8y - 4z = 16\]

Step 3: Add Equations to Eliminate the x-Term

Now add the new equivalent equation from Step 2 to the third equation:\[-4x + 5y + z = 10\]\[+ \]\[4x + 8y - 4z = 16\]By adding these equations, the \(x\)-terms cancel each other out, resulting in:\[(5y + z) + (8y - 4z) = 10 + 16\]\[13y - 3z = 26\]

Step 4: Write the New Equivalent System

The new system of equations after eliminating the \(x\)-term in the third equation is:\[\begin{aligned}2x - y + 3z &= 2 \x + 2y - z &= 4 \13y - 3z &= 26\end{aligned}\]

Key Concepts

Elimination Method

The elimination method is a powerful technique used in solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining ones. This method is especially useful when equations are already set up to allow one variable to be straightforwardly cancelled out.

Here's how it works:

• Identify the variable you want to eliminate from one of the equations.
• Choose another equation and manipulate it by multiplication to match the coefficient of the variable you wish to eliminate.
• Add or subtract the resulting equations to cancel out that variable.

In the provided solution, the elimination method was applied to the system by focusing on the elimination of the x-term from the third equation. By manipulating the second equation to have a matching coefficient for the x-term, these equations were combined to remove the x-term, simplifying the system. The elimination method is a versatile tool for solving linear equations, especially when dealing with systems involving three or more equations.

Equivalent Systems

In linear algebra, creating equivalent systems involves transforming one system into another that has the same solutions as the original. This is a critical aspect when using the elimination method.

Equivalent systems are useful because they often lead to simpler equations, making it easier to find solutions. Here's what's important about creating an equivalent system:

• Ensure any operations performed on equations preserve equality, such as adding or subtracting whole equations, or multiplying an entire equation by a non-zero constant.
• The solution set remains unchanged, meaning if you solve one system, the solutions should work in all of the equivalent systems.

In the given solution, the transformation of the second equation into a form that can be directly added to the third equation preserved the equality. The success of creating equivalent systems lies in maintaining the same solution set while performing transformations that simplify solving.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors and equations involving linear relationships. Mastery of linear algebra is crucial for effectively working with systems of linear equations.

This field encompasses key concepts such as:

• Vectors and vector spaces, providing a framework for understanding directions and magnitudes.
• Matrices, which can represent and simplify systems of linear equations for easier manipulation, including row operations.
• Transformations, allowing systems to be manipulated into equivalent, often simpler forms.

In the context of the textbook exercise, linear algebra principles govern the process of eliminating variables to arrive at simpler systems, ready for solving. By understanding and applying these fundamental principles, solving complex systems becomes more accessible and intuitive. Linear algebra thus offers a wide toolkit for dealing with linear relationships, critical for applications across mathematics and related fields.