Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} &\left(\frac{1}{2}, \frac{1}{2},-2\right)\\\ &3 x+y+z=0\\\ &4 x+2 y+z=1\\\ &2 x-2 y-z=2 \end{aligned}$$

Step-by-Step Solution

Answer

The ordered triple (\(\frac{1}{2}\), \(\frac{1}{2}\), \(-2\)) is a solution of the system.

Step 1: Understand the System of Equations

We have a system of three equations with three variables: \(x\), \(y\), and \(z\). The system is:1. \(3x + y + z = 0\)2. \(4x + 2y + z = 1\)3. \(2x - 2y - z = 2\).We are given the ordered triple \(\left(\frac{1}{2}, \frac{1}{2}, -2\right)\), which represents values for \(x\), \(y\), and \(z\), respectively.

Step 2: Substitute into the First Equation

Substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the first equation:\[3\left(\frac{1}{2}\right) + \frac{1}{2} - 2 = 0\].Calculate:\[\frac{3}{2} + \frac{1}{2} - 2 = \frac{4}{2} - 2 = 2 - 2 = 0\].The identity holds true.

Step 3: Substitute into the Second Equation

Use \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) in the second equation:\[4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 2 = 1\].Calculate:\[2 + 1 - 2 = 1\].The identity holds true as well.

Step 4: Substitute into the Third Equation

Substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the third equation:\[2\left(\frac{1}{2}\right) - 2\left(\frac{1}{2}\right) + 2 = 2\].Calculate:\[1 - 1 + 2 = 2\].This identity also holds true.

Key Concepts

Ordered Triple

An ordered triple is a set of three numbers written in a specific order, typically represented as \(x, y, z\). These numbers specify the exact values for the variables in a system of equations. In the context of this problem, the ordered triple \(\frac{1}{2}, \frac{1}{2}, -2\) represents the values for the variables \(x\), \(y\), and \(z\), respectively. These numbers are plugged into the equations to determine if they satisfy all parts of the system.
Understanding the ordered triple is essential because it serves as the potential solution to the system of equations. Each position in the triple has a distinct role: \

• First value corresponds to \(x\).
• Second value corresponds to \(y\).
• Third value corresponds to \(z\).

In this exercise, the ordered triple is given, and we need to verify its correctness by plugging it into each equation of the system. This gives us insight into how the ordered values interact with each equation.

Substitution

Substitution is a method used to verify solutions in a system of equations by replacing the variables with the given values from an ordered triple. In other words, you take the numbers provided in the ordered triple and substitute them for the variables in each equation. This is a crucial step in verifying whether the ordered triple is indeed a valid solution.
In our example:

• We substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the first equation to become \[3(\frac{1}{2}) + \frac{1}{2} - 2 = 0\].
• Perform similar substitutions for the second and third equations.

By performing substitution, we transform each equation into a simple arithmetic expression. This transformation allows for easy checking if each equation holds true (i.e., if each side of the equation equals after substituting the values). If all equations are satisfied, the ordered triple is a correct solution.

Identity Verification

Identity verification involves confirming whether the substitutions result in all parts of the system satisfying the given equations. This process is simple yet essential as it ensures that the substituted values yield true identities — the left-hand side of each equation balances with the right-hand side.
For example, in the first equation:

• After substituting the values, calculate to find out whether each side of the equation is equal. \[3(\frac{1}{2}) + \frac{1}{2} - 2\] simplifies to \[0\].

The exercise does this for all three equations in the system.

• For the second equation, when you substitute and simplify, you should get \[1\] on both sides.
• For the third equation, substitution leads to \[2\], confirming the identity for that equation.

Successfully verifying each identity confirms the ordered triple as a valid solution to the system. Not only does this ensure correctness, but it's also a method to detect any potential errors in problem-solving.