Use the echelon method to solve the following system of two equations in two unknowns. Check your answer.
6 x plus 3 y
equalsnegative 12
5 x minus 3 y
equalsnegative 21


We are given the system of linear equations:

\[
\begin{aligned}
6x + 3y &= -12 \\
5x - 3y &= -21
\end{aligned}
\]

To solve using the echelon method, we will follow these steps:

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

\[
\begin{pmatrix}
6 & 3 & | & -12 \\
5 & -3 & | & -21
\end{pmatrix}
\]

### Step 2: Perform row operations to reach echelon form

We want the matrix in the form where we can easily back-substitute. First, let's make the leading coefficient of the first row 1 by dividing the entire first row by 6.

\[
R_1 = \frac{1}{6} R_1
\]

This gives us:

\[
\begin{pmatrix}
1 & \frac{1}{2} & | & -2 \\
5 & -3 & | & -21
\end{pmatrix}
\]

Next, we'll eliminate the \(5\) in the second row by subtracting \(5 \times R_1\) from \(R_2\):

\[
R_2 = R_2 - 5R_1
\]

This gives us:

\[
\begin{pmatrix}
1 & \frac{1}{2} & | & -2 \\
0 & -\frac{11}{2} & | & -11
\end{pmatrix}
\]

### Step 3: Solve the system using back-substitution

From the second row, we can solve for \(y\):

\[
-\frac{11}{2} y = -11
\]

Multiplying both sides by \(-2/11\):

\[
y = 2
\]

Now substitute \(y = 2\) into the first row:

\[
1x + \frac{1}{2}(2) = -2
\]

Simplifying:

\[
x + 1 = -2 \implies x = -3
\]

### Step 4: Check the solution

Substitute \(x = -3\) and \(y = 2\) into the original equations:

\[
6(-3) + 3(2) = -18 + 6 = -12 \quad \text{(True)}
\]
\[
5(-3) - 3(2) = -15 - 6 = -21 \quad \text{(True)}
\]

Thus, the solution is correct.

### Final Answer:
\[
x = -3, \quad y = 2
\]

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

### Related Questions:
1. What is the general procedure for solving a system of equations using the echelon method?
2. How do you determine if a system of equations has no solution using row echelon form?
3. Can the echelon method be used for non-square systems of equations?
4. How does back-substitution work in Gaussian elimination?
5. What are the differences between Gaussian elimination and Gauss-Jordan elimination?

### Tip:
When using the echelon method, always aim to have leading 1's in your rows, as this simplifies back-substitution.

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer.
6x + 3y = -12
5x - 3y = -21

Learn how to solve the system of equations 6x + 3y = -12 and 5x - 3y = -21 using the echelon method. Follow step-by-step instructions, from forming the augmented matrix to applying row operations and back-substitution. Suitable for Grades 9-12.

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