Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 2 x+y-z &=-4 \\ y+2 z &=12 \\ 2 x-z &=-4 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \((x, y, z) = (1, 0, 6)\). The equations are independent, with a unique solution.

Step 1: Identify the equations

We are given the following system of equations:1. \( 2x + y - z = -4 \)2. \( y + 2z = 12 \)3. \( 2x - z = -4 \) We'll solve this system to find if the equations are dependent and determine the solution set.

Step 2: Eliminate a variable

First, let's eliminate \(x\) by subtracting equation (3) from equation (1):\[(2x + y - z) - (2x - z) = -4 - (-4)\]This simplifies to:\[y = 0\]Now, substitute \(y = 0\) into equation (2):\[0 + 2z = 12\]Which simplifies to:\[z = 6\]

Step 3: Solve for the last variable

Substitute \(z = 6\) into equation (3) to solve for \(x\):\[2x - 6 = -4\]Adding 6 to both sides gives:\[2x = 2\]Divide by 2:\[x = 1\]

Step 4: Verify the solution

Substitute \(x = 1\), \(y = 0\), and \(z = 6\) back into each equation:- For equation (1): \(2(1) + 0 - 6 = -4\) (True)- For equation (2): \(0 + 2(6) = 12\) (True)- For equation (3): \(2(1) - 6 = -4\) (True)All equations are satisfied, confirming the solution.

Key Concepts

Analytical Methods

When solving systems of equations, one approach is using analytical methods. These are precise algebraic techniques that help find solutions without having to use numerical approximations.
One common analytical method is the elimination method, which involves combining equations to eliminate one of the variables, as shown in the original exercise. This method is particularly useful when dealing with linear systems.
Another is substitution, where you solve one of the equations for one variable and then substitute this into the other equations. This requires careful manipulation of equations to ensure accuracy in the solution.

Dependent Equations

In a system of equations, dependent equations are those that do not provide new information. Essentially, they are multiple representations of the same line or plane in the context of the problem.
Such systems might have infinite solutions, typically expressed in terms of a parameter, such as the variable \(z\) in the exercise. However, in our case, the given system is consistent and independent, meaning each equation contributes unique information.
Recognizing dependent equations can simplify the solving process by reducing the number of necessary computations. By doing this, you avoid redundant steps and focus on the equations that truly define the solution set.

Solution Verification

After finding a potential solution to a system of equations, it is essential to verify that the solution satisfies all the equations. This step ensures the correctness of the solution and confirms that no mistakes were made during calculation.
In the exercise, the solution was verified by substituting \(x = 1\), \(y = 0\), and \(z = 6\) back into each original equation.

• Equation (1): Affirmation through substitution ensures no arithmetic errors.
• Equation (2): Verifying against all possible scenarios to confirm consistency.
• Equation (3): Checks that the manipulated definitions match the original conditions.

This triple-check guarantees the solution meets all criteria and conditions set by the equations.

Substitution Method

Substitution is a cornerstone technique in solving systems of equations. It involves solving one equation for one of the variables, and then using this expression to replace the same variable in the other equations.
This can simplify systems significantly, especially when one equation is already close to being defined in terms of a single variable.
In the step-by-step solution, after eliminating \(x\), we used substitution by plugging \(y = 0\) into the second equation to find \(z\), and subsequently into the third equation to determine \(x\).

• This method helps isolate variables systematically.
• It reduces the complexity of multi-variable algebraic equations.
• Substitution ensures each variable is accurately and sequentially determined.

Understanding how to apply this method is crucial for efficiently solving not only simple but also more complex equation systems.