Problem 6 - College Algebra with Modeling and Visualization

Question

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{array}{l} (5,2,2),(2,-1,1) \\ 2 x-3 y+3 z=10 \\ x-2 y-3 z=1 \\ 4 x-y+z=10 \end{array} $$

Step-by-Step Solution

Answer

The ordered triple \((5, 2, 2)\) is not a solution; \((2, -1, 1)\) is a solution.

Step 1: Introduce the Problem

We need to determine if the ordered triples \((5,2,2)\) and \((2,-1,1)\) are solutions to the system of linear equations given by: \(2x-3y+3z=10\), \(x-2y-3z=1\), and \(4x-y+z=10\). This means we need to substitute each of the triples into the equations and verify if they satisfy all the equations simultaneously.

Step 2: Check Ordered Triple (5, 2, 2)

Substitute \(x=5\), \(y=2\), \(z=2\) into each equation:- First equation: \(2(5) - 3(2) + 3(2) = 10\). Simplifying gives: \(10 - 6 + 6 = 10\), which is true.- Second equation: \(5 - 2(2) - 3(2) = 1\). Simplifying gives: \(5 - 4 - 6 = -5\), which is false.- Third equation: \(4(5) - 2 + 2 = 10\). Simplifying gives: \(20 - 2 + 2 = 20\), which is false.

Step 3: Conclude on Triple (5, 2, 2)

Since the ordered triple \((5, 2, 2)\) does not satisfy all of the equations (it fails the second and third equations), it is not a solution to the system of equations.

Step 4: Check Ordered Triple (2, -1, 1)

Substitute \(x=2\), \(y=-1\), \(z=1\) into each equation:- First equation: \(2(2) - 3(-1) + 3(1) = 10\). Simplifying gives: \(4 + 3 + 3 = 10\), which is true.- Second equation: \(2 - 2(-1) - 3(1) = 1\). Simplifying gives: \(2 + 2 - 3 = 1\), which is true.- Third equation: \(4(2) - (-1) + 1 = 10\). Simplifying gives: \(8 + 1 + 1 = 10\), which is true.

Step 5: Conclude on Triple (2, -1, 1)

Since the ordered triple \((2, -1, 1)\) satisfies all of the equations, it is a solution to the system of equations.

Key Concepts

Ordered Triples

An ordered triple is a group of three numbers, usually written in parentheses, that represent coordinates in three-dimensional space. For instance, an ordered triple like \(5, 2, 2\) identifies a point in space where \(x = 5\), \(y = 2\), and \(z = 2\). In the context of solving systems of linear equations, each ordered triple is a potential solution that needs to satisfy all of the equations in the system.

• They serve as possible candidates to test against the equations.
• If an ordered triple makes each equation true when the values are substituted, it is a solution of the system.

In our exercise, we are testing to see whether the ordered triples \(5, 2, 2\) and \(2, -1, 1\) meet the criteria for being solutions to the system of equations.

Substitution Method

The substitution method is a technique used to check if potential solutions satisfy a set of equations. It involves replacing the variables in the equations with the proposed values from the ordered triple:

• Begin by substituting each individual value into each equation where those variables appear.
• Simplicity can help determine whether the equation holds true or false.

Let's take an example: for the ordered triple \(5, 2, 2\):- Substitute \(x = 5\), \(y = 2\), and \(z = 2\) into the equation \(2x - 3y + 3z = 10\). Then we calculate: \[2(5) - 3(2) + 3(2) = 10\] This simplifies correctly to \(10 - 6 + 6 = 10\), which means the first equation is satisfied.Using the substitution method systematically verifies whether all combinations are solutions. This method is straightforward and works well, especially when dealing with simple numbers and equations.

Solution Verification

Solution verification is the process of confirming if a set of values meets the requirements of each equation in the system. After we have substituted the values, we need to:

• Check each result to ensure it satisfies all the equations.
• Hence, confirm whether or not the values form a legitimate solution.

For instance, using the triple \(2, -1, 1\):- Begin by checking \(2x - 3y + 3z = 10\): \[2(2) - 3(-1) + 3(1) = 10\] Simplifies accurately to \(4 + 3 + 3 = 10\), thus, confirming as true.Verification is crucial to ensure accuracy because it's entirely possible for values to satisfy some equations while failing others, as was the case with the first triple \(5, 2, 2\). By this approach, only values confirmed through verification can be trusted as true solutions.