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Chapter 7: Problem 6

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 2 x-6 y+6 z &=-12 \\ x+4 y+5 z &=-1 \quad \text { and }(0,1,-1) \\\\-x+2 y+3 z &=-1 \end{aligned} $$

Short Answer

Expert verified
Yes, \((0, 1, -1)\) is a solution to the system.

Step by step solution

01

Understand the Problem

We need to determine if the ordered triple \((0,1,-1)\) is a solution to the given system of equations. This means we must substitute \(x = 0\), \(y = 1\), and \(z = -1\) into each of the equations and check if the equations are true.
02

Substitute into the First Equation

Substitute \((0, 1, -1)\) into the first equation: \(2x - 6y + 6z = -12\).Substitute the values: \(2(0) - 6(1) + 6(-1) = 0 - 6 - 6 = -12\). The left-hand side is \(-12\), which equals the right-hand side. So the solution satisfies the first equation.
03

Substitute into the Second Equation

Substitute \((0, 1, -1)\) into the second equation: \(x + 4y + 5z = -1\).Substitute the values: \(0 + 4(1) + 5(-1) = 0 + 4 - 5 = -1\). The left-hand side is \(-1\), which equals the right-hand side. So the solution satisfies the second equation.
04

Substitute into the Third Equation

Substitute \((0, 1, -1)\) into the third equation: \(-x + 2y + 3z = -1\).Substitute the values: \(-0 + 2(1) + 3(-1) = 0 + 2 - 3 = -1\). The left-hand side is \(-1\), which equals the right-hand side. So the solution satisfies the third equation.
05

Conclusion: Verify the Solution

After substituting \((0, 1, -1)\) into all three equations, we confirmed that it satisfies each equation. Therefore, the ordered triple \((0, 1, -1)\) is the solution to the system of equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Triples
An ordered triple is a set of three numbers written in a specific order, like \( (x, y, z) \). These numbers represent solutions to systems of three equations. Think of them like an address in a 3D space where each number gives a different coordinate.
In our exercise, the ordered triple we are checking is \( (0, 1, -1) \). This means we substitute \( x = 0 \), \( y = 1 \), and \( z = -1 \) into each of the provided equations.
  • It is vital that all three numbers meet the conditions set by all equations of the system to qualify as a valid solution.
  • If even one part fails, the triple is not valid for that system.
Substitution Method
The substitution method involves solving systems of equations by replacing variables with their corresponding values from the ordered triple.We substitute the values of \( x, y, \) and \( z \) into the equations one by one.
For example, in the first equation, \( 2x - 6y + 6z = -12 \), you replace \( x = 0 \), \( y = 1 \), and \( z = -1 \) and calculate to see if the left-hand side equals the right-hand side.
  • This method is useful for verifying if an ordered triple precisely solves each equation in the system.
  • It's like a check step-by-step if the proposed solution works for every part of the system.
By substituting into all equations and confirming the results, we verify the integrity of the solution.
Solution Verification
Solution verification is the process of confirming whether a proposed solution truly satisfies every part of a system of equations.This involves substituting the values back into the original equations as outlined in the substitution step.
If each equation ends up being true with the substituted values, then the solution is verified.In our exercise, after substitution, each equation was satisfied by the ordered triple.
  • This means that \( (0, 1, -1) \) is indeed the correct solution for the entire system.
  • Verification helps prevent errors by ensuring no steps were missed during the solving process.
Verified solutions instill confidence that the mathematical problem-solving process was accurate and thorough.

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Most popular questions from this chapter

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