{−2x−15y=9−2x−14y=4 Use elementary row operations on the augmented matrix in part (a) to make a matrix of the form

We are given the system of linear equations:

\[
\begin{aligned}
-2x - 15y &= 9 \\
-2x - 14y &= 4
\end{aligned}
\]

The goal is to use elementary row operations to transform the augmented matrix into a simplified form. First, let’s represent the system as an augmented matrix.

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

\[
\begin{pmatrix}
-2 & -15 & 9 \\
-2 & -14 & 4
\end{pmatrix}
\]

### Step 2: Apply Row Operations

To simplify the matrix, we will perform elementary row operations. Here’s the plan:

1. Make the coefficient of \(x\) in the first row equal to 1.
2. Use that row to eliminate the \(x\)-coefficient from the second row.
3. Continue simplifying until the matrix is in row-echelon form (the desired matrix form).

#### Step 2.1: Row 1 - Divide by -2

Divide Row 1 by -2 to make the first element a 1:

\[
R1 \rightarrow \frac{R1}{-2}
\]

This gives:

\[
\begin{pmatrix}
1 & \frac{15}{2} & -\frac{9}{2} \\
-2 & -14 & 4
\end{pmatrix}
\]

#### Step 2.2: Row 2 - Add Row 1 multiplied by 2 to Row 2

Now, eliminate the \(x\)-term from the second row by performing the operation:

\[
R2 \rightarrow R2 + 2 \cdot R1
\]

\[
R2 = \begin{pmatrix} -2 & -14 & 4 \end{pmatrix} + 2 \cdot \begin{pmatrix} 1 & \frac{15}{2} & -\frac{9}{2} \end{pmatrix}
\]

Calculating this, we get:

\[
R2 = \begin{pmatrix} 0 & 1 & -5 \end{pmatrix}
\]

Thus, the matrix becomes:

\[
\begin{pmatrix}
1 & \frac{15}{2} & -\frac{9}{2} \\
0 & 1 & -5
\end{pmatrix}
\]

### Step 3: Simplify

Now that the matrix is in an upper triangular form (row-echelon form), this is the desired result. The matrix represents the simplified system:

\[
\begin{aligned}
x + \frac{15}{2} y &= -\frac{9}{2} \\
y &= -5
\end{aligned}
\]

### Step 4: Solve the System

From the second equation, we can directly see that \(y = -5\). Substituting \(y = -5\) into the first equation:

\[
x + \frac{15}{2}(-5) = -\frac{9}{2}
\]

\[
x - \frac{75}{2} = -\frac{9}{2}
\]

\[
x = -\frac{9}{2} + \frac{75}{2} = \frac{66}{2} = 33
\]

So, the solution is \(x = 33\) and \(y = -5\).

### Conclusion:
The system has a unique solution: \(x = 33\) and \(y = -5\).

Would you like further details on any of the steps? 

Here are five related questions to consider:

1. How can you apply Gaussian elimination to larger systems of equations?
2. What are the different types of elementary row operations?
3. Can you solve a system if the matrix has no unique solution?
4. How can you use row-reduction to find the inverse of a matrix?
5. What is the difference between row-echelon form and reduced row-echelon form?

**Tip:** Always check the determinant of a matrix when solving systems—it can indicate whether the system has a unique solution!

-2x - 15y = 9, -2x - 14y = 4. Use elementary row operations on the augmented matrix in part (a) to make a matrix of the form.

Learn how to solve a system of linear equations using elementary row operations. This problem involves transforming an augmented matrix into row-echelon form to solve for x and y. Step-by-step instructions provided for Grades 10-12.

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