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:

  1. 7x6y=557x - 6y = -55
  2. 14x+12y=10-14x + 12y = 10

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!

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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