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Let's solve the system of equations using the echelon method:

The given system is:

\[
4x - 3y = -1 \tag{1}
\]
\[
-8x + 6y = 2 \tag{2}
\]

### Step 1: Create an augmented matrix for the system:
\[
\begin{pmatrix}
4 & -3 & | & -1 \\
-8 & 6 & | & 2
\end{pmatrix}
\]

### Step 2: Apply row operations to achieve echelon form.

First, we can divide the first row by 4 to make the leading coefficient of \(x\) in the first equation equal to 1:
\[
\begin{pmatrix}
1 & -\frac{3}{4} & | & -\frac{1}{4} \\
-8 & 6 & | & 2
\end{pmatrix}
\]

Next, we'll eliminate the \(x\)-term from the second row by adding 8 times the first row to the second row:
\[
R_2 = R_2 + 8R_1
\]

This gives us:
\[
\begin{pmatrix}
1 & -\frac{3}{4} & | & -\frac{1}{4} \\
0 & 0 & | & 0
\end{pmatrix}
\]

### Step 3: Interpret the echelon form.

The second row indicates \(0 = 0\), which is always true. Therefore, the system does not provide any new information from the second equation.

The first row represents:
\[
x - \frac{3}{4}y = -\frac{1}{4}
\]
which can be rewritten as:
\[
x = \frac{3}{4}y - \frac{1}{4}
\]

### Step 4: Solution.

Since the second row is trivially true, there are infinitely many solutions. The solution can be expressed in parametric form as:
\[
x = \frac{3}{4}y - \frac{1}{4}, \quad y = y
\]
where \(y\) is any real number.

Thus, the answer is **option B**: There are infinitely many solutions, and the general solution is \((\frac{3}{4}y - \frac{1}{4}, y)\), where \(y\) is any real number.

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Do you have any questions or need further details? Here are 5 related questions:

1. What is the significance of echelon form in solving systems of equations?
2. How do you identify when a system of linear equations has no solution?
3. How can you express the solution of a system with infinitely many solutions parametrically?
4. What other methods can be used to solve systems of linear equations besides the echelon method?
5. Why does a row of zeros in a matrix correspond to an identity like \(0 = 0\)?

**Tip:** When dealing with systems of equations, always check if you can simplify one of the equations to make row operations easier.

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer: 4x - 3y = -1, -8x + 6y = 2.

Learn how to solve the system of linear equations 4x - 3y = -1 and -8x + 6y = 2 using the echelon method. This detailed solution includes key algebraic concepts like augmented matrices and row operations, suitable for students in Grades 10-12.

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