Math Problem Statement
Show that the system of linear equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.
Solution
The system of linear equations in your image is as follows:
1. \quad x + y + z &= 6 \\ 2. \quad x + 2y + 3z &= 14 \\ 3. \quad x + 4y + 7z &= 30 \\ \end{aligned}$$ We need to determine whether this system is consistent, meaning whether it has a solution, and then solve it if it is. ### Step 1: Write the system in matrix form The augmented matrix corresponding to the system is: $$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 14 \\ 1 & 4 & 7 & 30 \end{array}\right]$$ ### Step 2: Perform row reduction We will now use Gaussian elimination to reduce the matrix to row-echelon form. 1. Subtract the first row from the second and third rows: $$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 8 \\ 0 & 3 & 6 & 24 \end{array}\right]$$ 2. Subtract 3 times the second row from the third row: $$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 8 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ The third row is all zeros, indicating that one of the equations is dependent on the others. Therefore, the system is consistent and has infinitely many solutions. ### Step 3: Solve the system We can express the system in terms of one of the variables, say $$z$$. From the second row: $$y + 2z = 8 \quad \Rightarrow \quad y = 8 - 2z$$ From the first row: $$x + y + z = 6 \quad \Rightarrow \quad x + (8 - 2z) + z = 6 \quad \Rightarrow \quad x = -2 + z$$ Thus, the general solution is: $$x = -2 + z, \quad y = 8 - 2z, \quad z = z$$ where $$z$$ is a free parameter. ### Conclusion: The system is consistent and has infinitely many solutions, with the solution set given by: $$\boxed{x = -2 + z, \quad y = 8 - 2z, \quad z = z}$$ where $$z$$ is any real number. Would you like further clarification or details on any part of the solution? ### Related Questions: 1. What are the conditions for a system of linear equations to be consistent? 2. How can you interpret a free parameter in a system of equations? 3. What is the difference between consistent and inconsistent systems? 4. How does Gaussian elimination help in solving systems of equations? 5. Can a system of equations have exactly one solution, and how can you tell? ### Tip: When solving systems of linear equations, always check for dependent equations by row-reducing the augmented matrix to see if one equation can be derived from others.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Matrix Operations
Formulas
Gaussian elimination
Row reduction for solving systems of equations
Theorems
Consistency of Linear Systems
Infinite Solutions in Linear Systems
Suitable Grade Level
Grades 10-12
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