Problem 3 - College Algebra

Question

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(4,1,2)\\\ &\left\\{\begin{aligned} x-2 y &=2 \\ 2 x+3 y &=11 \\ y-4 z &=-7 \end{aligned}\right. \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the ordered triple (4,1,2) is a solution to the given system of equations.

Step 1: Identify the given ordered triple and system of equations

The ordered triple given is (4,1,2). This means that \(x = 4\), \(y = 1\), and \(z = 2\). The system of equations to be verified is:\(x - 2y = 2\) \(2x + 3y = 11\) \(y - 4z = -7\)

Step 2: Substitute ordered triple values into the first equation

Place the values into the first equation: \(x - 2y = 2\) becomes \(4 - 2*1 = 2\). Simplifying that we get \(4 - 2 = 2\) which is a true statement. So, the coordinates satisfy the first equation.

Step 3: Substitute ordered triple values into the second equation

Next, place the values into the second equation: \(2x + 3y = 11\) becomes \(2*4 + 3*1 = 11\). Simplifying that we get \(8 + 3 = 11\) which is also a true statement. Thereby, the coordinates satisfy the second equation too.

Step 4: Substitute ordered triple values into the third equation

Lastly, place the values into the third equation: \(y - 4z = -7\) becomes \(1 - 4*2 = -7\). Simplifying that we get \(1 - 8 = -7\) which is a true statement yet again. Hence, the coordinates satisfy the third equation as well.

Key Concepts

Ordered Triple

When working with a system of linear equations involving three variables, one might come across the term "ordered triple." It is simply a set of three numbers that represent potential solutions. In this context, the ordered triple is usually denoted as \((x, y, z)\). Each number in the ordered triple corresponds to one of the variables in the system:

• \(x\): The first number, which represents the value assigned to the variable \(x\).
• \(y\): The second number, which represents the assigned value to the variable \(y\).
• \(z\): The third number, representing the value for the variable \(z\).

For instance, the ordered triple \((4, 1, 2)\) indicates \(x = 4\), \(y = 1\), and \(z = 2\). Substituting these numbers into the equations of the system helps us determine if this particular set of values is a solution to the system.
It's like a code that we use to solve a puzzle, checking if every piece fits perfectly into its place by satisfying all parts of the system.

Substitution Method

The substitution method is a technique used to determine whether a given ordered triple satisfies a system of linear equations. This method involves substituting the values of the variables from the ordered triple into each equation of the system.

Here's a brief breakdown of how it works:

• Identify each equation in the system and substitute the respective values of \(x\), \(y\), and \(z\) from the ordered triple.
• Simplify the expressions. For example, if the equation is \(x - 2y = 2\) and the ordered triple is \((4, 1, 2)\), replace \(x\) with 4 and \(y\) with 1: \(4 - 2(1) = 2\).
• Confirm whether the left-hand side of the equation equals the right-hand side after substitution."

This method is consistent, reliable, and often simpler than other techniques. It provides a step-by-step approach that breaks down the complex process into manageable pieces. By checking whether each equation holds true using the specified values, we verify the correctness of the ordered triple.

Solution Verification

Solution verification is the process of checking if an ordered triple truly solves a system of equations. After using the substitution method to plug the values into every equation, verifying involves making sure each equation is a true statement.

Here's how verification works:

• Substitute each variable in every equation with the values from the ordered triple.
• Simplify each equation to confirm both sides are equal, thereby ensuring that the equation holds true.
• If the computed verifications of all the equations in the system are true, then the ordered triple is indeed a solution to the system.

This step is essential, because it confirms our calculations and assumptions. It’s a reflection of disciplined problem-solving.
Just remember, one false equation means the ordered triple is not a solution. Ensuring accuracy in this step can avoid pitfalls in solving complex systems, and gives a clear picture of where errors might occur if solutions don't check out.