Solve The Given System Of Equations. If The System Has No Solution, Say That It Is Inconsistent.{x2y+3z=2x+y+z=3x+2y2z=17418Select
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Introduction
Solving systems of equations is a fundamental skill in algebra that allows us to find the values of variables that satisfy multiple equations simultaneously. When dealing with systems involving two or more variables, it becomes crucial to understand whether the system is consistent (has solutions) or inconsistent (has no solutions). This article provides a comprehensive guide on how to solve a given system of equations, analyze its consistency, and interpret the results.
In particular, we will examine the following system:
\[
\begin{cases}
x + 2y + 3z = 2 \\
x + y + z = 3 \\
3x + 2y + 2z = 17418
\end{cases}
\]
(Note: The original problem statement appears to contain formatting errors or typos, such as missing operators or inconsistent notation. For clarity, the equations are reconstructed based on logical interpretation.)
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Understanding the System of Equations
Before solving, let's understand the structure of the system:
- Equation 1: \( x + 2y + 3z = 2 \)
- Equation 2: \( x + y + z = 3 \)
- Equation 3: \( 3x + 2y + 2z = 17418 \)
This is a system of three equations with three variables: \(x, y, z\). Our goal is to find the values of \(x, y, z\) that satisfy all three equations simultaneously.
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Step-by-Step Solution Approach
Step 1: Write the system in matrix form
The system can be expressed as:
\[
\begin{bmatrix}
1 & 2 & 3 \\
1 & 1 & 1 \\
3 & 2 & 2
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
2 \\
3 \\
17418
\end{bmatrix}
\]
Step 2: Use elimination or substitution methods
To solve the system, we typically choose elimination or substitution. Here, elimination is effective.
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Solving the System
Step 3: Subtract Equation 2 from Equation 1
Equation 1: \( x + 2y + 3z = 2 \)
Equation 2: \( x + y + z = 3 \)
Subtracting Eq. 2 from Eq. 1:
\[
(x + 2y + 3z) - (x + y + z) = 2 - 3
\]
\[
( x - x ) + ( 2y - y ) + ( 3z - z ) = -1
\]
\[
0 + y + 2z = -1
\]
This simplifies to:
\[
y + 2z = -1 \quad \text{(Equation 4)}
\]
Step 4: Use Equation 2 to express \(x\)
From Equation 2:
\[
x + y + z = 3
\]
\[
x = 3 - y - z
\]
Step 5: Substitute \(x\) into Equation 3
Equation 3:
\[
3x + 2y + 2z = 17418
\]
Substituting \(x = 3 - y - z\):
\[
3(3 - y - z) + 2y + 2z = 17418
\]
\[
9 - 3y - 3z + 2y + 2z = 17418
\]
\[
9 - y - z = 17418
\]
Rearranged:
\[
-y - z = 17418 - 9
\]
\[
-y - z = 17409
\]
Multiply both sides by -1:
\[
y + z = -17409 \quad \text{(Equation 5)}
\]
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Step 6: Solve the system with Equations 4 and 5
Recall:
- Equation 4: \( y + 2z = -1 \)
- Equation 5: \( y + z = -17409 \)
Subtract Equation 5 from Equation 4:
\[
(y + 2z) - (y + z) = -1 - (-17409)
\]
\[
( y - y ) + ( 2z - z ) = 17408
\]
\[
0 + z = 17408
\]
Thus,
\[
z = 17408
\]
Step 7: Find \( y \)
From Equation 5:
\[
y + z = -17409
\]
Substitute \(z = 17408\):
\[
y + 17408 = -17409
\]
\[
y = -17409 - 17408
\]
\[
y = -34817
\]
Step 8: Find \( x \)
Recall:
\[
x = 3 - y - z
\]
Substitute \( y = -34817 \) and \( z = 17408 \):
\[
x = 3 - (-34817) - 17408
\]
\[
x = 3 + 34817 - 17408
\]
\[
x = (3 + 34817) - 17408
\]
\[
x = 34820 - 17408
\]
\[
x = 17412
\]
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Final Solution:
\[
\boxed{
\begin{cases}
x = 17412 \\
y = -34817 \\
z = 17408
\end{cases}
}
\]
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Verifying the Solution
It's important to verify that these values satisfy all original equations.
Equation 1:
\[
x + 2y + 3z = 17412 + 2(-34817) + 3(17408)
\]
\[
= 17412 - 69634 + 52224
\]
\[
= (17412 + 52224) - 69634
\]
\[
= 69636 - 69634 = 2
\]
Correct.
Equation 2:
\[
x + y + z = 17412 - 34817 + 17408
\]
\[
= (17412 + 17408) - 34817
\]
\[
= 34820 - 34817 = 3
\]
Correct.
Equation 3:
\[
3x + 2y + 2z = 3(17412) + 2(-34817) + 2(17408)
\]
\[
= 52236 - 69634 + 34816
\]
\[
= (52236 + 34816) - 69634
\]
\[
= 87052 - 69634 = 17418
\]
Correct.
Thus, the solution satisfies all equations.
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Is the System Consistent?
Since we found a unique solution that satisfies all three equations, the system is consistent. The solution is unique.
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Additional Insights and Tips for Solving Systems of Equations
- Recognize the Type of System
- Consistent and Independent: Has a unique solution.
- Consistent and Dependent: Infinite solutions (e.g., equations represent the same plane).
- Inconsistent: No solution exists, system is incompatible.
- Methods for Solving Systems
- Substitution: Best when one variable can be easily isolated.
- Elimination: Effective for eliminating variables systematically.
- Matrix Methods: Gaussian elimination or using matrices and determinants for larger systems.
- Dealing with Large or Complex Systems
- Use calculators or software tools like MATLAB, WolframAlpha, or Python libraries (NumPy, SymPy) to handle computations.
- Always verify solutions by substituting back into original equations.
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When Does a System Have No Solution?
A system is inconsistent if, after applying elimination or substitution, you arrive at a contradiction—such as \(0 = 5\)—indicating no set of variable values satisfies all equations simultaneously. For example:
\[
\begin{cases}
x + y = 2 \\
x + y = 3
\end{cases}
\]
These are inconsistent because they imply different values for \(x + y\).
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Summary
- We successfully solved the given system and found the unique solution:
\[
\boxed{
x = 17412, \quad y = -34817, \quad z = 17408
}
\]
- The system is consistent since a solution exists.
- Verifying solutions in the original equations confirms correctness.
- Understanding the nature of systems (consistent or inconsistent) is essential in algebra and helps in analyzing real-world problems involving multiple conditions.
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Final Thoughts
Mastering techniques to solve systems of equations is a valuable skill in mathematics, engineering, and sciences. Whether using substitution, elimination, or matrix methods, always verify the solutions and understand whether the system is consistent or inconsistent. In cases where no solutions are found, clearly state that the system is inconsistent,
