Use the echelon method to solve the following system of two equations in two unknowns. Check your answer.

2 x plus 32x+3y

equals=negative 2−2

3 x minus 2 y3x−2y

equals=1010

Let's first write the system of equations clearly:

1. \( 2x + 3y = -2 \)
2. \( 3x - 2y = 10 \)

We will use the echelon method (Gaussian elimination) to solve this system.

### Step 1: Write the system in augmented matrix form

\[
\begin{pmatrix}
2 & 3 & | & -2 \\
3 & -2 & | & 10
\end{pmatrix}
\]

### Step 2: Eliminate the first element in the second row

We want to make the coefficient of \(x\) in the second row equal to zero. We can do this by multiplying the first row by \( \frac{3}{2} \) and subtracting it from the second row.

Multiply row 1 by \( \frac{3}{2} \):

\[
\left( \frac{3}{2} \times \begin{pmatrix}
2 & 3 & | & -2
\end{pmatrix} \right) = \begin{pmatrix}
3 & 4.5 & | & -3
\end{pmatrix}
\]

Now subtract this from row 2:

\[
\begin{pmatrix}
3 & -2 & | & 10
\end{pmatrix} - \begin{pmatrix}
3 & 4.5 & | & -3
\end{pmatrix} = \begin{pmatrix}
0 & -6.5 & | & 13
\end{pmatrix}
\]

So the new augmented matrix is:

\[
\begin{pmatrix}
2 & 3 & | & -2 \\
0 & -6.5 & | & 13
\end{pmatrix}
\]

### Step 3: Solve for \(y\)

From the second row, we have the equation:

\[
-6.5y = 13
\]

Solving for \(y\):

\[
y = \frac{13}{-6.5} = -2
\]

### Step 4: Back-substitute to find \(x\)

Now that we know \(y = -2\), substitute this value into the first equation to solve for \(x\):

\[
2x + 3(-2) = -2
\]

\[
2x - 6 = -2
\]

\[
2x = 4
\]

\[
x = 2
\]

### Step 5: Check the solution

Substitute \(x = 2\) and \(y = -2\) into the second original equation:

\[
3(2) - 2(-2) = 10
\]

\[
6 + 4 = 10
\]

The solution \(x = 2\) and \(y = -2\) satisfies both equations.

Thus, the solution to the system is:
\[
x = 2, \quad y = -2
\]

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How can you represent a system of equations using matrices?
2. What is the difference between row echelon form and reduced row echelon form?
3. Can the echelon method be used for systems with more than two equations?
4. What if one of the rows becomes entirely zero during Gaussian elimination?
5. How would you interpret the solution geometrically?

**Tip**: Always check your solution by substituting it back into both original equations.

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer: 2x + 3y = -2 and 3x - 2y = 10

Learn how to solve a system of linear equations using the echelon method (Gaussian elimination). This problem involves solving the system 2x + 3y = -2 and 3x - 2y = 10 by transforming it into an augmented matrix, followed by elimination and back-substitution. Suitable for high school students in Grades 9-12.

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