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
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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.

Let's solve the system of equations using the echelon method. The two equations are:

1. \( 7x - 6y = -55 \)
2. \( -14x + 12y = 10 \)

### Step 1: Write the system as an augmented matrix.

The system can be represented as the following augmented matrix:
\[
\begin{pmatrix}
7 & -6 & | & -55 \\
-14 & 12 & | & 10
\end{pmatrix}
\]

### 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.**

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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!

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer. 7x − 6y = −55 and −14x + 12y = 10.

Learn how to solve a system of linear equations using the echelon method. This example includes the equations 7x − 6y = −55 and −14x + 12y = 10. Discover step-by-step how to transform the system into echelon form, and check for solution consistency. Suitable for students in Grades 9-12.

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