Problem 6 - A Graphical Approach to College Algebra

Question

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \begin{aligned} &\text\\\\\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

Yes, \((\frac{1}{2}, \frac{1}{2}, -2)\) is a solution.

Step 1: Substitute Values into the First Equation

Start with the first equation: \(3x + y + z = 0\). Substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the equation. This gives:\[3\left(\frac{1}{2}\right) + \frac{1}{2} - 2 = 0\]Simplify the expression:\[\frac{3}{2} + \frac{1}{2} - 2 = 0\]\[2 - 2 = 0\]Therefore, the first equation is satisfied.

Step 2: Substitute Values into the Second Equation

Move on to the second equation: \(4x + 2y + z = 1\). Substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the equation. This gives:\[4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 2 = 1\]Simplify:\[2 + 1 - 2 = 1\]\[1 = 1\]Therefore, the second equation is satisfied.

Step 3: Substitute Values into the Third Equation

Proceed to the third equation: \(2x - 2y - z = 2\). Substitute \(x = \frac{1}{2}\), \(y = \frac{1}{2}\), and \(z = -2\) into the equation. This gives:\[2\left(\frac{1}{2}\right) - 2\left(\frac{1}{2}\right) + 2 = 2\]Simplify:\[1 - 1 + 2 = 2\]\[2 = 2\]Thus, the third equation is satisfied.

Step 4: Conclusion

Since the values satisfy all three equations in the system, the ordered triple \(\left(\frac{1}{2}, \frac{1}{2}, -2\right)\) is indeed a solution to the system of equations.

Key Concepts

Understanding an Ordered Triple

An ordered triple consists of three numbers organized in a specific order:

• The first position typically represents the value of \(x\).
• The second position represents the value of \(y\).
• The third position represents the value of \(z\).

In our example, the given ordered triple is \(\left(\frac{1}{2}, \frac{1}{2}, -2\right)\).
Think of an ordered triple as the solution's potential values that need to satisfy each equation when placed correctly.
This concept extends the idea of ordered pairs used for two-variable equations to systems involving three variables, adding an extra layer of mathematical expression.

Essential Steps for Solution Verification

Verification of a solution involves substituting each value of the ordered triple into the involved equations.
It aims to ensure that all equations are satisfied, confirming the integrity of the solution.

• Substitute: Replace the variables \(x\), \(y\), and \(z\) in each equation with the values from the ordered triple.
• Simplify: Perform the operations required, following mathematical rules to simplify each equation.
• Compare: Once simplified, each side of the equation should be equal. If true for all, the solution is verified.

By going through these steps, we ensure all equations hold true with the given values.
This builds confidence in the solution's correctness and helps catch potential errors if some equations don't match initially.

The Substitution Method

The substitution method is a fundamental technique used to solve systems of equations.
In this context, it is applied to demonstrate the correctness of an ordered triple. Here's how substitution works:

• Identify an equation that is easy to manipulate.
• Use one variable to express the others, if needed, or simply substitute values directly from the ordered triple as in our example.
• Plug these values into another equation to explore dependencies and verify results.

The beauty of substitution lies in its simplicity—it allows you to solve and verify systems systematically and thoroughly.
Through stepwise replacement, it not only verifies a given triple but, in other scenarios, can help find the correct solution by simplifying the problem step-by-step.