Problem 9 - College Algebra

Question

Determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} x-y &=0 \\ x-z &=5 \quad \text { and }(4,4,-1) \\ x-y+z &=-1 \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the ordered triple \((4, 4, -1)\) is the solution to the system of equations.

Step 1: Insert values into the first equation

The first equation is \(x-y = 0\). Insert \(x = 4\) and \(y = 4\) into it. We get \(4 - 4 = 0\), which simplifies to \(0 = 0\). Thus, the first equation holds true with this ordered triple.

Step 2: Insert values into the second equation

The second equation is \(x-z = 5\). Insert \(x = 4\) and \(z = -1\) into it. We get \(4 - (-1) = 4 + 1 = 5\), which simplifies to \(5 = 5\). Thus, the second equation holds true with this ordered triple.

Step 3: Insert values into the third equation

The third equation is \(x-y+z = -1\). Insert \(x = 4\), \(y = 4\), and \(z = -1\) into it. We get \(4 - 4 + (-1) = 4 - 4 - 1 = -1\), which simplifies to \(-1 = -1\). Thus, the third equation holds true with this ordered triple.

Step 4: Conclusion

Since the ordered triple \((4, 4, -1)\) makes all three equations true, it is indeed the solution to the system of equations.

Key Concepts

Understanding Ordered Triples

In algebra, when dealing with systems of equations that involve three variables, we often encounter the term "ordered triples." An ordered triple is a set of three numbers, written in a specific order, which corresponds to the values of the variables in equations. For example, in the ordered triple \((4, 4, -1)\), **the first number (4)** is the value for **\(x\), the second number (4)** is the value for **\(y\), and the third number (-1)** is the value for **\(z\).**

Ordered triples are crucial when solving systems of equations because they help verify if a given combination of values satisfies all equations within the system. Each equation in a system needs to be true for the ordered triple to be a solution. If even one equation is not satisfied, then the ordered triple is not a solution.

Substitution Method

The substitution method is a popular technique used to solve systems of equations. This method involves solving one equation for one variable and then substituting this expression into the other equations. Although not explicitly used in this exercise, understanding substitution can enhance comprehension of systems.

Here's a simplified example: Suppose you have the system \(x + y = 6\) and \(y - 3x = 4\). Solving the first equation for **\(x\)** gives \(x = 6 - y\). **Now you substitute \(6 - y\) for \(x\)** in the second equation. This gives \(y - 3(6 - y) = 4\), which simplifies and allows you to solve for **\(y\)**. Once **\(y\)** is known, you can substitute back to find **\(x\)**.

This strategy highlights how using substitution can simplify solving complex systems of equations.

Algebraic Solutions in Systems of Equations

Algebraic solutions involve manipulating equations using basic algebraic operations like addition, subtraction, multiplication, and division to find values for variables. In the context of systems of equations, we solve for each variable algebraically by ensuring all original equations hold true with the found values.

To employ algebraic solutions:

• Identify and isolate variables using given equations.
• Substitute known values into other equations to simplify them.
• Check if these simplified equations hold true with the proposed values.

In the exercise provided, each of the three equations was proven true by inserting the values from the ordered triple \((4, 4, -1)\). Such clarity with substitution prevents mistakes or overlooking pivotal solutions in complex systems.

Solving Equations Effectively

Solving equations effectively requires a systematic approach to ensure accuracy and efficiency. Here are some strategies:

• Break complex problems into smaller parts. Focus on one equation at a time.
• Use substitution or elimination methods as needed.
• Ensure each calculation is verified by double-checking the work.
• If possible, graphically represent solutions to get a visual understanding.

By solving each equation separately and verifying with known solutions, like the ordered triple in the given problem, you confirm the accuracy of your approach. Consistency in checking solutions against each equation helps solidify understanding and prevents errors.