Math Problem Statement
Use the echelon method to solve the following system of two equations in two unknowns. Check your answer. 7 x minus 6 y7x−6y equals=minus−55 negative 14 x plus 12 y−14x+12y equals=1010 Question content area bottom Part 1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is enter your response here. (Type an ordered pair.) B. There are infinitely many solutions. The general solution is left parenthesis nothing comma y right parenthesisenter your response here,y, where y is any real number. C. There is no solution.
Solution
Let's solve the system of equations using the echelon method. The two equations are:
Step 1: Write the system as an augmented matrix.
The system can be represented as the following augmented matrix:
7 & -6 & | & -55 \\ -14 & 12 & | & 10 \end{pmatrix}$$ ### Step 2: Perform row operations to get the matrix into echelon form. #### Row operation 1: We will replace the second row by adding $$2 \times \text{Row 1}$$ to $$\text{Row 2}$$, in order to eliminate the $$x$$-term from the second row. - $$\text{Row 2} = \text{Row 2} + 2 \times \text{Row 1}$$ $$\begin{pmatrix} 7 & -6 & | & -55 \\ (-14 + 14) & (12 - 12) & | & (10 - 110) \end{pmatrix}$$ $$\begin{pmatrix} 7 & -6 & | & -55 \\ 0 & 0 & | & -100 \end{pmatrix}$$ ### Step 3: Analyze the echelon form. The second row becomes $$0x + 0y = -100$$, which simplifies to $$0 = -100$$. This is a contradiction. ### Conclusion: Since we have a contradiction, the system has no solution. Thus, the correct answer is: **C. There is no solution.** --- Would you like further clarification or more details? Here are some related questions to practice: 1. How would you use substitution to solve the same system of equations? 2. What is the difference between inconsistent and dependent systems? 3. How can you determine if a system has infinitely many solutions? 4. Can a system of linear equations have exactly two solutions? 5. How do row operations help in simplifying a system of equations? **Tip:** Always check for consistency when you encounter a row with all zeros on the left-hand side but a nonzero constant on the right. This signals no solution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Echelon Form
Formulas
Row Operations
Echelon Form of a Matrix
Theorems
Inconsistency in Linear Systems
Suitable Grade Level
Grades 9-12
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