Problem 11 - Algebra and Trigonometry

Question

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x-2 y-z &=4 \\ x-y+3 z &=0 \\ 2 x+y+z &=0 \end{aligned}\right. $$ Eliminate the \(x\) -term from the second equation.

Step-by-Step Solution

Answer

The new equivalent system is: \(x - 2y - z = 4\), \(y + 4z = 0\), and \(2x + y + z = 0\).

Step 1: Analyze the System

The given system of equations is: 1) \(x - 2y - z = 4\) 2) \(x - y + 3z = 0\) 3) \(2x + y + z = 0\). Our task is to eliminate the \(x\)-term from the second equation.

Step 2: Subtract First Equation from Second

To eliminate the \(x\)-term from the second equation, subtract the first equation from the second. Let's rewrite the equations for clarity: \((x - y + 3z) - (x - 2y - z)\).

Step 3: Simplify to Create New Equation

Now simplify the result from Step 2: \(x - y + 3z - x + 2y + z = 0\). This reduces to \(y + 4z = 0\).

Step 4: Write New System

The new system of equations after eliminating \(x\) from the second equation is: 1) \(x - 2y - z = 4\) 2) \(y + 4z = 0\) 3) \(2x + y + z = 0\).

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. It involves the strategic elimination of variables by adding or subtracting equations from each other. This method helps in reducing the system to one that is easier to solve. To use the elimination method effectively, follow these steps:

• Identify the variable you want to eliminate from one or more of the equations.
• Adjust coefficients if necessary, by multiplying, dividing, or scaling one or both equations, so that adding or subtracting the equations will eliminate the selected variable.
• Perform the arithmetic operation (addition or subtraction) to get a new equation with the reduced number of variables.

In this specific exercise, the aim was to remove the \(x\)-term from the second equation to simplify the system. By subtracting the first equation from the second, the \(x\) variable was effectively eliminated, leading to a simpler expression for the equation.

Solving Linear Equations

Once you've successfully eliminated a variable using methods like elimination, you may need to solve the linear equations that result from this process. Solving linear equations typically involves further simplifications. Here are a few key steps:

• Rearrange the equation so that all terms involving the variables are on one side and constant terms are on the other.
• Isolate the variable of interest by performing operations such as addition, subtraction, multiplication, or division on both sides of the equation.
• Check your solution by substituting it back into the original equations to verify that it satisfies all parts of the system.

In our practice problem, after eliminating \(x\), we are left with simpler equations such as \(y + 4z = 0\). This equation can now be solved for one variable in terms of the others, aiding further resolution of the overall system.

Equivalent Systems

Understanding equivalent systems is crucial in solving systems of equations efficiently. Two systems of equations are considered equivalent if they have exactly the same solutions. This concept allows us to transform a complex system into a simpler one without changing the solution set.

The operations involved in creating equivalent systems include:

• Adding or subtracting a multiple of one equation to another, like what was done in the elimination step.
• Multiplying or dividing an entire equation by a nonzero constant. This doesn't change the solutions but can simplify the numbers you're working with.

In our exercise, after eliminating a variable, the new system of equations \(x - 2y - z = 4\), \(y + 4z = 0\), and \(2x + y + z = 0\) is equivalent to the original. The solutions that satisfy the first system will also solve the new system, illustrating the effectiveness of creating equivalent systems in solving equations.