Problem 3 - Beginning and Intermediate Algebra

Question

Determine if the ordered triple is a solution of the system. \(-x+y-2 z=2\) \(3 x-y+5 z=4\) \(2 x+3 y-z=7\) \((0,6,2)\)

Step-by-Step Solution

Answer

The ordered triple (0, 6, 2) satisfies equations 1 and 2 but not equation 3 in the given system. Therefore, it is not a solution to the entire system of equations.

Step 1: Write down the given system of equations and ordered triple

The given system of equations is: \[ \begin{cases} -x + y - 2z = 2 \\ 3x - y + 5z = 4 \\ 2x + 3y - z = 7 \end{cases} \] And the given ordered triple is: (0, 6, 2)

Step 2: Substitute the values of the variables in each equation

We need to substitute the values of x, y, and z in each equation with the given ordered triple values. Therefore for (x, y, z) = (0, 6, 2), we have: Equation 1: \(- (0) + (6) - 2(2) = 2\) Equation 2: \(3 (0) - (6) + 5(2) = 4\) Equation 3: \(2 (0) + 3(6) - (2) = 7\)

Step 3: Perform the operations and check if the equations are true

Now, let's perform the operations in each equation and check if the result on the left-hand side equals the right-hand side. Equation 1: \[- (0) + (6) - 2(2) = -0 + 6 - 4 = 2\] The result matches the right-hand side, so the ordered triple satisfies equation 1. Equation 2: \[3 (0) - (6) + 5(2) = 0 - 6 + 10 = 4\] The result matches the right-hand side, so the ordered triple satisfies equation 2. Equation 3: \[2 (0) + 3(6) - (2) = 0 + 18 - 2 = 16\] The result does not match the right-hand side, which means the ordered triple does not satisfy equation 3.

Step 4: Conclusion

Since the ordered triple (0, 6, 2) satisfies equations 1 and 2 but not equation 3, it's not a solution to the entire system of equations.

Key Concepts

Ordered Triple

An ordered triple is an extension of the idea of an ordered pair into three dimensions. It consists of a set of three numbers, written in a specific order, and usually represented as \(x, y, z\). These numbers correspond to the values in three-dimensional space, typically representing the coordinate values along the three axes.

• The order is crucial: \( (1, 2, 3) \) is not the same as \( (3, 2, 1) \).
• Ordered triples are used especially in systems of linear equations with three variables.

This concept is fundamental when determining if a particular set of values solves a system of equations. You substitute each number of the triple into the respective variables (x, y, z) in your equations to see if they hold true.

In our case, the ordered triple \( (0, 6, 2) \) needs to be checked against a system of three equations to see if it satisfies all of them.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves replacing one variable with an expression that results from another equation, simplifying the problem into smaller, more manageable parts. This method can be very useful in confirming whether an ordered triple is a solution to the system of equations.

To use substitution here, follow these steps:

• First, choose any equation to substitute the ordered values for x, y, and z.
• For each equation in the system, replace the variables with corresponding values from the ordered triple.

The goal of substitution in this context is to see if each equation balances, or in other words, whether the left-hand side equals the right-hand side. Such a substitution helps in easily verifying each of the equations individually for the same ordered triple.

Equation Verification

Equation verification is the process of confirming whether or not a proposed solution satisfies each equation in a system. This involves a straightforward substitution and calculation method, where you replace each variable in the equation with their respective values from the ordered triple.

• Begin by substituting the values into every equation in the system.
• Perform all operations on the left-hand side to simplify and determine the result.
• Compare the outcome with the right-hand side of the equation to confirm if the equation holds true.

If all equations are satisfied (true), the ordered triple is a solution to the system. However, as illustrated in our example, if even one equation fails (as equation 3 does when using the values from \(0, 6, 2\)), then the entire ordered triple cannot be considered a solution to the system.

Solving Linear Equations

Solving linear equations is a core activity in algebra that involves finding the values of variables that make the equation true. When a system of linear equations involves three variables, solving involves finding an ordered triple which satisfies all the given equations.

• Each linear equation can be thought of as a plane in three-dimensional space.
• The solution is the set of points where these planes intersect.

In simpler terms, for a set of three linear equations, a particular ordered triple needs to satisfy all three equations to be a solution.
This requires extensive practice in algebraic manipulation and conceptual understanding of spatial relationships. By solving such systems, students enhance their critical-thinking and problem-solving skills significantly.