Problem 12 - Algebra and Trigonometry

Question

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x+y-3 z &=3 \\\\-2 x+3 y+z &=2 \\ x-y+2 z &=0 \end{aligned}\right. $$ Eliminate the \(x\) -term from the second equation

Step-by-Step Solution

Answer

The new equivalent system is: \(\{\begin{aligned} x+y-3z &=3\\ 5y-5z &=8\\ x-y+2z &=0 \end{aligned}\}\.\)

Step 1: Identify the Coefficient of x

The first step is to identify the coefficient of \(x\) in the first and second equations. In the first equation, the coefficient is 1, and in the second equation, it is -2.

Step 2: Create the Opposite Coefficient for x

To eliminate \(x\) from the second equation, we multiply the first equation by 2, so that the \(x\)-terms in both equations have opposite coefficients. The first equation becomes:\[ 2(x+y-3z) = 2(3) \]which simplifies to:\[ 2x + 2y - 6z = 6 \]

Step 3: Add the Equations

Add the new version of the first equation to the second equation to eliminate \(x\):\[\begin{aligned}2x + 2y - 6z &= 6 \-2x + 3y + z &= 2 \\end{aligned}\]Adding these gives:\[ (2x - 2x) + (2y + 3y) + (-6z + z) = 6 + 2 \]This simplifies to:\[ 5y - 5z = 8 \]

Step 4: Write the New System

With the \(x\)-term eliminated from the second equation, the new equivalent system of equations is:\[\begin{aligned}x + y - 3z &= 3 \5y - 5z &= 8 \x - y + 2z &= 0 \\end{aligned}\]

Key Concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. This method involves eliminating one variable at a time by combining equations, making it easier to solve for the remaining variables. By strategically adding or subtracting entire equations, you can isolate a variable, reducing the complexity of the system.

Here's a simple breakdown of how it works:

• Identify the coefficients of the variable you want to eliminate in both equations.
• Modify the equations so that these coefficients are equal but opposite in sign.
• Add or subtract the equations to eliminate the chosen variable.

In our case, we multiplied the entire first equation by 2, so that the coefficient of \(x\) in the first equation (2) cancels out with the coefficient of \(x\) in the second equation (-2). This results in a new system where the \(x\) variable is effectively removed from the second equation.

Linear Equations

Linear equations form the foundation of the elimination method, and understanding them is crucial. A linear equation is any equation that, when graphed, forms a straight line. It typically takes the form of \(ax + by + cz = d\), where \(x\), \(y\), and \(z\) are variables and \(a\), \(b\), \(c\), and \(d\) are constants.

Key characteristics of linear equations include:

• The highest power of the variables is 1.
• There are no products or roots of variables.
• Graphical representation is a straight line.

In our exercise, each equation, like \(x + y - 3z = 3\), is linear in nature, ensuring that the elimination method can be applied effectively to find a solution.

Systems of Equations

A system of equations comprises multiple equations that are simultaneously true. These systems can have one solution (a point where all lines intersect), no solution (when lines are parallel), or infinitely many solutions (when lines overlap perfectly).

To solve a system of equations, you need to find values for the variables that satisfy all equations in the system. The elimination method is one of several techniques used to achieve this. Its advantage lies in systematically reducing the number of variables by combining equations.

For our exercise, the goal was to manipulate the given system using the elimination method to derive a simpler equivalent system. By eliminating \(x\) from the second equation, we clarified the relationships between \(y\) and \(z\), paving the way to solving the system with fewer hurdles.