Math Problem Statement
For what values of a the system of equation is consistent
Solution
To determine the values of for which the given system of equations is consistent, we must analyze the following system:
x - y + z = 1 \\ x + 2y + 4z = a \\ x + 4y + 6z = a_2 \end{cases}$$ ### Step 1: Write the System in Matrix Form The augmented matrix for this system is: $$\begin{bmatrix} 1 & -1 & 1 & | & 1 \\ 1 & 2 & 4 & | & a \\ 1 & 4 & 6 & | & a_2 \\ \end{bmatrix}$$ ### Step 2: Perform Row Operations 1. **Subtract Row 1 from Row 2 and Row 3** to make the first column zeroes below the first row. This leads to: $$\begin{bmatrix} 1 & -1 & 1 & | & 1 \\ 0 & 3 & 3 & | & a - 1 \\ 0 & 5 & 5 & | & a_2 - 1 \\ \end{bmatrix}$$ 2. **Subtract Row 2 from Row 3** to simplify further: $$\begin{bmatrix} 1 & -1 & 1 & | & 1 \\ 0 & 3 & 3 & | & a - 1 \\ 0 & 0 & 0 & | & a_2 - a \\ \end{bmatrix}$$ ### Step 3: Consistency Condition For the system to be consistent, the last row $$[0, 0, 0 | a_2 - a]$$ implies that the system will have a solution if: $$a_2 - a = 0 \Rightarrow a_2 = a$$ ### Conclusion The system is consistent if $$a_2 = a$$. Would you like more details on these steps, or do you have any specific questions? --- ### Related Questions 1. How can we determine if a system has infinitely many solutions? 2. What does it mean for a system of equations to be inconsistent? 3. How can we solve for $$x$$, $$y$$, and $$z$$ when $$a = a_2$$? 4. What other methods can be used to check the consistency of a system? 5. How would the solution change if the equations were non-linear? ### Tip In general, to check the consistency of a system, focus on the row echelon form to see if any contradictions (like a row of zeroes equaling a non-zero constant) arise.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Consistency
Formulas
Row reduction to row echelon form
Theorems
Consistency condition for a system of linear equations
Suitable Grade Level
Grades 11-12 or early college
Related Recommendation
Finding the Parameter a for No Solutions in a System of Equations
Determine Parameter 'a' for Unique, Infinite, or No Solutions in a System of Linear Equations
Determine the Value of b for Consistency in a Linear System Using Gaussian Elimination
Consistency of a Linear System Using Gaussian Elimination
Solving Systems of Linear Equations with Parameters a, b, c for Unique, Infinite, or No Solutions