Problem 2 - 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. $$\begin{aligned} \left(\frac{1}{2},-\frac{3}{4}, \frac{1}{6}\right) & \\ 2 x+8 y-6 z &=-6 \\ x+y+z &=-\frac{1}{12} \\ x+3 z &=1 \end{aligned}$$

Step-by-Step Solution

Answer

Yes, the ordered triple is a solution to the system.

Step 1: Substituting in the First Equation

Substitute the given ordered triple \( \left( \frac{1}{2}, -\frac{3}{4}, \frac{1}{6} \right) \) into the first equation \( 2x + 8y - 6z = -6 \). We have:\[2 \left( \frac{1}{2} \right) + 8 \left( -\frac{3}{4} \right) - 6 \left( \frac{1}{6} \right) = -6\]Calculate each term: - \(2 \times \frac{1}{2} = 1\)- \(8 \times -\frac{3}{4} = -6\)- \(6 \times \frac{1}{6} = 1\)Substitute back: \[1 - 6 - 1 = -6\]The left side equals the right side, so it satisfies the first equation.

Step 2: Substituting in the Second Equation

Next, substitute the given ordered triple \( \left( \frac{1}{2}, -\frac{3}{4}, \frac{1}{6} \right) \) into the second equation \( x + y + z = -\frac{1}{12} \). We have:\[\frac{1}{2} + \left( -\frac{3}{4} \right) + \frac{1}{6} = -\frac{1}{12}\]Convert to a common denominator (use 12):- \(\frac{1}{2} = \frac{6}{12}\)- \(-\frac{3}{4} = -\frac{9}{12}\)- \(\frac{1}{6} = \frac{2}{12}\)Combine:\[\frac{6}{12} - \frac{9}{12} + \frac{2}{12} = \frac{-1}{12}\]The left side equals the right side, so it satisfies the second equation.

Step 3: Substituting in the Third Equation

Finally, substitute the given ordered triple \( \left( \frac{1}{2}, -\frac{3}{4}, \frac{1}{6} \right) \) into the third equation \( x + 3z = 1 \). We have:\[\frac{1}{2} + 3 \times \frac{1}{6} = 1\]Calculate the term:- \(3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\)Combine:\[\frac{1}{2} + \frac{1}{2} = 1\]The left side equals the right side, so it satisfies the third equation.

Key Concepts

Ordered Triples

Ordered triples are a fundamental concept in the study of linear systems. An ordered triple consists of three values, each corresponding to a particular variable in a three-dimensional space, usually denoted as \( (x, y, z) \). In the context of solving systems of linear equations, an ordered triple represents a specific solution that satisfies all equations in the system. To verify an ordered triple, we substitute these values into each equation and check that they hold true.
For example, the ordered triple \( \left( \frac{1}{2}, -\frac{3}{4}, \frac{1}{6} \right) \) translates into substituting \( x = \frac{1}{2}\), \( y = -\frac{3}{4}\), and \( z = \frac{1}{6} \) in our given system of equations. A correct substitution that satisfies all equations demonstrates that the ordered triple is indeed a solution.

Solution Verification

Solution verification is the process of confirming that a proposed solution actually satisfies the equations of a system. In this case, we substitute each component of the given ordered triple back into the original equations one by one.

• First, substitute into the first equation: \( 2x + 8y - 6z = -6 \). If the left-hand side evaluates to \(-6\) after substitution, as shown, it confirms correctness for that equation.


• Next, check the second equation: \( x + y + z = -\frac{1}{12} \). This step involves carefully adding fractions and finding a common denominator to ensure that both sides equal \(-\frac{1}{12}\).


• Finally, verify the third equation: \( x + 3z = 1 \). Accurate arithmetic with fractions confirms that the left side equals the right side, thus verifying the solution for this equation as well.

Each substitution must hold to consider the ordered triple a valid solution for the entire system.

Fractional Coefficients

Fractional coefficients add complexity to solving and verifying solutions of linear systems. They require careful arithmetic to avoid errors and ensure solutions are correct.
Familiarity with fractional arithmetic is crucial—converting fractions to a common denominator, multiplying fractions, and simplification are frequent operations needed.

• In the exercise, each equation contains fractions that need careful handling to maintain correctness. For example, turning \( \frac{1}{2} \) into \( \frac{6}{12} \) helps align it with \( \frac{9}{12} \) from \(-\frac{3}{4}\) when summing for the second equation.


• Understanding how to manage fractions also extends to operations such as multiplication, evident in the first and third equations where terms like \( 6 \times \frac{1}{6} \) simplify to whole numbers or new fractions.

Precision with fractions ensures that verification is accurate, and the ordered triple genuinely solves the system of equations.