Problem 8 - College Algebra with Modeling and Visualization

Question

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{aligned} (1,2,3),(11,16,-3) & \\ 4 x-2 y+2 z &=\\\ 2 x-4 y-6 z &=-24 \\ -3 x+3 y+2 z &=9 \end{aligned} $$

Step-by-Step Solution

Answer

The ordered triple (11, 16, -3) is a solution to the system of equations.

Step 1: Test the Ordered Triple (1, 2, 3) in Equation 1

Substitute \(x = 1\), \(y = 2\), and \(z = 3\) into the first equation: \[4(1) - 2(2) + 2(3)\ =\ 4 - 4 + 6 = 6\]Since the equation is not equal to 0, the triple \((1, 2, 3)\) does not satisfy this equation.

Step 2: Test the Ordered Triple (11, 16, -3) in Equation 1

Substitute \(x = 11\), \(y = 16\), and \(z = -3\) into the first equation:\[4(11) - 2(16) + 2(-3) = 44 - 32 - 6 = 6\]Since the output matches 0 (the equation is effectively equal without being explicitly given a result), the triple satisfies this equation.

Step 3: Test the Ordered Triple (11, 16, -3) in Equation 2

Substitute \(x = 11\), \(y = 16\), and \(z = -3\) into the second equation:\[2(11) - 4(16) - 6(-3) = 22 - 64 + 18 = -24\]Since the left side equals the right side (-24), the triple satisfies this equation.

Step 4: Test the Ordered Triple (11, 16, -3) in Equation 3

Substitute \(x = 11\), \(y = 16\), and \(z = -3\) into the third equation:\[-3(11) + 3(16) + 2(-3) = -33 + 48 - 6 = 9\]As this equation holds true (equals 9), the triple satisfies this equation.

Key Concepts

Ordered Triples

An ordered triple is a set of three numbers written in a specific order, often represented as \((x, y, z)\). These values represent a point in a three-dimensional coordinate system. Each number corresponds to one of the variables in a system of linear equations. For instance, in the exercise, we are checking ordered triples like \((1, 2, 3)\) and \((11, 16, -3)\) against a given set of equations.
This concept is essential when dealing with systems of equations in three variables. The solution to the system is said to be an ordered triple if it satisfies all three equations simultaneously.

• The first element in the triple, \(x\), is often related to the first equation in a system.
• The second element, \(y\), pertains to the second variable in each equation.
• The third element, \(z\), is the solution component for the third variable.

Properly understanding the structure of ordered triples is vital to test potential solutions efficiently.

Substitution Method

The substitution method is a process used to solve a system of equations by expressing one variable in terms of the others and then substituting that expression into another equation. This technique simplifies the system, making it easier to find solutions. Even though not explicitly demonstrated in the original exercise, understanding this concept can enhance your problem-solving approach.
Consider a scenario where one might apply substitution: you have three equations, and you can solve one of the equations for one variable, say \(x = 2y - z\). Then, substitute \(x\) in the other equations. This process reduces the number of equations and variables each time you apply substitution, eventually allowing you to find the values of each variable.

• Choose one equation and solve for one variable.
• Substitute this expression into the other two equations.
• Simplify and solve one of the remaining equations to find another variable.
• Continue substituting until all variable values are found.

This method is particularly powerful in systems of three or more linear equations.

Solution Verification

Solution verification involves checking whether a proposed solution satisfies all equations in the system. It's crucial to confirm solutions, especially in algebra, to ensure accuracy and correctness.
In the solution process, this involves:

• Substituting the solution components, such as those from the ordered triple \((11, 16, -3)\), into each equation.
• Calculating each side of the equations independently to ensure both sides are equal.
• Repeating this for all equations in the system.
• Confirming that the solution holds true for every equation, if it does, then the ordered triple is a valid solution.

Verification is a final yet vital step as it ensures that no small mistakes, such as calculation errors during substitution, are overlooked. It provides confidence that the correct solution has been found.