Problem 10 - College Algebra

Question

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{array}{l}{-x-y+2 z=3} \\ {5 x+8 y-3 z=4 \quad \text { and }(4,1,-7)} \\\ {-x+3 y-5 z=-5}\end{array} $$

Step-by-Step Solution

Answer

The ordered triple (4, 1, -7) is not a solution to the system of equations.

Step 1: Understand the Problem

We need to determine if the ordered triple \((4, 1, -7)\) is a solution to the given system of equations by substituting the values into each equation and checking if the equations hold true.

Step 2: Substitute into the First Equation

Substitute \(x = 4\), \(y = 1\), and \(z = -7\) into the first equation: \(-x - y + 2z = 3\). Calculate: \(-4 - 1 + 2(-7)\). This simplifies to \(-4 - 1 - 14 = -19\). This is not equal to \(3\).

Step 3: Conclusion

Since the result from substituting into the first equation did not satisfy the equation, \((4, 1, -7)\) is not a solution to the system of equations. There is no need to check further equations because at least one equation not being satisfied is sufficient to conclude that the ordered triple is not a solution.

Key Concepts

Understanding Ordered Triple

An ordered triple consists of three elements, typically written in the form \((x, y, z)\). These elements represent values from a system of three equations. They are 'ordered' because the position of each value is critical. The first position corresponds to \(x\), the second to \(y\), and the third to \(z\).
For example, the ordered triple \((4, 1, -7)\) indicates that \(x = 4\), \(y = 1\), and \(z = -7\). In a system of equations, the goal is to find an ordered triple such that each equation is true when these values are plugged in.

• Each ordered triple provides a potential solution to the equations.
• If substituting an ordered triple into any equation fails to hold true, then it is not a solution to the given system.

This principle emphasizes that solving a system requires finding an ordered triple that satisfies all equations simultaneously.

Exploring the Substitution Method

The substitution method is a fundamental technique used to determine if a certain ordered triple is a solution to a system of equations. This involves replacing the variables in the equations with values from the ordered triple.
When applying the substitution method, follow these steps:

• Take the first element from the ordered triple and substitute it for \(x\) in all the equations.
• Replace \(y\) with the second element, and \(z\) with the third.
• Simplify each equation to check if it holds true.

This method helps bridge the gap between abstract algebraic expressions and concrete numerical values, allowing for a practical check for solutions.

The Importance of Solution Verification

Solution verification is crucial in ensuring the accuracy of results obtained from a system of equations. It involves confirming that the ordered triple indeed satisfies each equation within the system.
The verification process is straightforward:

• Substitute the values of the ordered triple into each equation.
• Perform arithmetic calculations to see if the left-hand side of the equation equals the right-hand side.
• If the equality holds for all equations, the triple is a true solution; otherwise, it is not.

In the given exercise, verifying the triple \((4, 1, -7)\) resulted in one equation failing the test, indicating it was not a solution. This step is vital, as even one unsatisfied equation invalidates the entire ordered triple as a solution.

Understanding Algebraic Equations

Algebraic equations form the backbone of solving systems using methods like substitution. Each equation consists of variables and constants linked by operations like addition or multiplication.
In a system, equations work together to pinpoint a common solution that fits them all. For our system:

• \(-x - y + 2z = 3\)
• \(5x + 8y - 3z = 4\)
• \(-x + 3y - 5z = -5\)

Each one serves as a constraint. The task is finding values satisfying all simultaneously. In essence, solving these equations isn't about working on them in isolation but understanding how they interact to pinpoint a valid solution.