Solve the system below using augmented matrix methods.
4x1-3x2=-7
-12x1+9x2=21

To solve the system of equations using the augmented matrix method, follow these steps:

### System of Equations:
1. \( 4x_1 - 3x_2 = -7 \)
2. \( -12x_1 + 9x_2 = 21 \)

### Step 1: Write the Augmented Matrix
The augmented matrix for the system is:

\[
\begin{pmatrix}
4 & -3 & | & -7 \\
-12 & 9 & | & 21
\end{pmatrix}
\]

### Step 2: Row Operations
We aim to make this matrix into reduced row-echelon form (RREF).

#### Step 2.1: Normalize the first row
Let's normalize the first row by dividing it by 4 (to make the leading entry of the first row 1):

\[
R_1 \to \frac{R_1}{4}
\]

\[
\begin{pmatrix}
1 & -\frac{3}{4} & | & -\frac{7}{4} \\
-12 & 9 & | & 21
\end{pmatrix}
\]

#### Step 2.2: Eliminate the \(x_1\)-term from the second row
Now, to eliminate the \(x_1\)-term from the second row, perform the row operation:

\[
R_2 \to R_2 + 12R_1
\]

\[
\begin{pmatrix}
1 & -\frac{3}{4} & | & -\frac{7}{4} \\
0 & 0 & | & 0
\end{pmatrix}
\]

### Step 3: Interpret the Result
The second row becomes all zeros, which means the system is dependent. This implies there are infinitely many solutions. The system can be written as:

\[
x_1 - \frac{3}{4}x_2 = -\frac{7}{4}
\]

### Step 4: Express \(x_1\) in terms of \(x_2\)
Solve for \(x_1\):

\[
x_1 = \frac{3}{4}x_2 - \frac{7}{4}
\]

Thus, the solution is given by the parametric form:

\[
x_1 = \frac{3}{4}t - \frac{7}{4}, \quad x_2 = t
\]

where \(t\) is any real number.

### Conclusion:
The system has infinitely many solutions and can be expressed parametrically as:

\[
(x_1, x_2) = \left( \frac{3}{4}t - \frac{7}{4}, t \right) \quad \text{for any } t \in \mathbb{R}.
\]

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Do you have any questions or would you like further details?

Here are 5 related questions to expand your understanding:
1. What is the general condition for a system of equations to have infinitely many solutions?
2. How can you verify the solution to this system using substitution?
3. How does the rank of the matrix affect the number of solutions in a system of equations?
4. Can you describe how the augmented matrix method is different from Gaussian elimination?
5. How would the solution change if the second equation were slightly different?

**Tip:** When a system leads to a row of zeros, it usually indicates that the system is either dependent or inconsistent, depending on the augmented part of the matrix.

Explore how to solve the system of equations 4x1-3x2=-7 and -12x1+9x2=21 using the augmented matrix method. Follow a step-by-step solution that introduces key linear algebra concepts such as row reduction and parametric solutions. Suitable for high school students in Grades 9-12.

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