Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{rr}3 x-y+z= & 1 \\ 2 x-3 z= & -14 \\ 5 y+2 z= & 8\end{array}\right.\) (a) (3,5,-3) (b) (-1,0,4) (c) (0,-1,3) (d) (1,0,4)

Step-by-Step Solution

Answer

Upon plugging the ordered triples into the system of equations, only ordered triple (a) - (3,5,-3) is a solution.

Step 1: Analyze Ordered Triple (a) - (3,5,-3)

For the ordered triple (3,5,-3), substitute x=3, y=5, z=-3 into each equation. If each equality holds true, then (3,5,-3) is a solution. If not, it's not a solution.

Step 2: Analyze Ordered Triple (b) - (-1,0,4)

For the ordered triple (-1,0,4), substitute x=-1, y=0, z=4 into each equation. If each equality holds true, then (-1,0,4) is a solution. If not, it's not a solution.

Step 3: Analyze Ordered Triple (c) - (0,-1,3)

For the ordered triple (0,-1,3), substitute x=0, y=-1, z=3 into each equation. If each equality holds true, then (0,-1,3) is a solution. If not, it's not a solution.

Step 4: Analyze Ordered Triple (d) - (1,0,4)

For the ordered triple (1,0,4), substitute x=1, y=0, z=4 into each equation. If each equality holds true, then (1,0,4) is a solution. If not, it's not a solution.

Key Concepts

Understanding Ordered Triples

When dealing with systems of equations, an ordered triple is a set of three numbers, typically written in the form \((x, y, z)\). Each number in the ordered triple corresponds to one of the variables in the equations of the system.
These values are substituted into the equations to check if they satisfy all the equations simultaneously. For example, consider the ordered triple (3,5,-3).
In this case, 3 is the value for \(x\), 5 is the value for \(y\), and -3 is the value for \(z\). It’s crucial to make sure that they fulfill each equation in the system for them to be a solution.

• Helps decipher if a point is part of the solution set.
• Keeps the process organized by assigning specific values to equations.
• Promotes structured testing of potential solutions.

Solution Verification and Consistency

After determining potential solutions through ordered triples, the next critical step is solution verification. Solution verification is the process of testing these substitutions within the original equations to confirm their validity.
This can be done by plugging each value of an ordered triple into the system and checking if all equations are satisfied. If even one equation returns false, the ordered triple is not a solution.

Steps to Verify Solutions

1. Substitute each variable \(x\), \(y\), \(z\) in one equation at a time.
2. Simplify both sides of each equation.
3. Confirm if the left hand side equals the right hand side for all equations.

• Ensures all equations agree with the given values.
• Provides conclusive evidence towards solution possibilities.
• Establishes consistency and coherence in the system.

Substitution Method in Equation Solving

The substitution method is a frequently used technique for solving systems of equations, particularly useful in handling ordered triples. It involves substituting one equation’s expression into another to reduce the number of variables, progressively simplifying the system.
In our context, substitution helps us insert values from the ordered triple into the corresponding equations. By doing this, we can verify if a particular ordered triple also satisfies all requirements of the system.

Why Use Substitution?

• Makes handling multi-variable systems more manageable.
• Can effectively simplify complex systems by reducing variables.
• Provides a systematic approach to solution discovery.

Understanding how the substitution method works equips students with a powerful tool not only for verification but also for developing further strategies in solving different kinds of equations.