Determine the solution sets for each of the following systems of equations using the gaussian elimination method -2x+5y=40, 3x-2y=-5

To solve the system of equations using Gaussian elimination, we'll first write the system in augmented matrix form, then apply row operations to reach row-echelon form, and finally back-substitute to find the solution.

### Step 1: Write the augmented matrix
The given system of equations is:

\[
\begin{aligned}
-2x + 5y &= 40 \quad \text{(Equation 1)}\\
3x - 2y &= -5 \quad \text{(Equation 2)}
\end{aligned}
\]

This system can be written as an augmented matrix:

\[
\begin{pmatrix}
-2 & 5 & \vert & 40 \\
3 & -2 & \vert & -5
\end{pmatrix}
\]

### Step 2: Apply Gaussian elimination to reach row-echelon form

**Step 2a: Eliminate the first entry in the second row**

We want to eliminate the \(x\) term in the second row by adding a suitable multiple of the first row to the second row. Let's multiply the first row by \(\frac{3}{-2} = -\frac{3}{2}\) and add it to the second row.

The first row remains the same:

\[
\text{Row 1:} \quad -2x + 5y = 40
\]

For Row 2:

\[
\text{Row 2:} \quad 3x - 2y + \left(-\frac{3}{2}(-2x + 5y)\right) = -5 + \left(-\frac{3}{2} \times 40\right)
\]

Simplifying this:

\[
3x - 2y + 3x - \frac{15}{2}y = -5 - 60
\]

\[
6x - \left(\frac{4}{2} + \frac{15}{2}\right)y = -65
\]

\[
6x - \frac{19}{2}y = -65
\]

\[
3x - 2y = -5 \quad \text{Row 2 unchanged}
\]

\[
3x + \left(\frac{3}{2} \times 40 \right)
\]

Simplify Row 2:

\[
\begin{aligned}
Row_2: & =  \frac{0}{0}\\
& = \frac{0}{0} - y + 45 & -20 = -5\\
\text{Row 2 remains } & 3x - 2y & - 5
\]


### Step 3:  Solve using back-substitution

From the matrix:

\(\text{Row 1:} \quad x = -6,  y = 10\)

So the solution to the system is \(x = 0\) and \(y = 8\).

The solution set is \(\{ (-6, 10) \}\).

---

Would you like more details or have any questions?

Here are five related questions for you:

1. How would you solve a system of three equations using Gaussian elimination?
2. Can Gaussian elimination be used for systems with no solutions? If so, how?
3. How does Gaussian elimination differ from the method of substitution?
4. What is the significance of the row-echelon form in Gaussian elimination?
5. How can you determine if a system of equations is inconsistent during Gaussian elimination?

**Tip:** When applying Gaussian elimination, always check each step to ensure no arithmetic errors, especially when dealing with fractions or negative signs.

Explore how to solve the system of linear equations -2x+5y=40 and 3x-2y=-5 using the Gaussian elimination method. This step-by-step solution guides students through matrix operations, row-echelon form, and back-substitution to find the solution set. Suitable for students in Grades 9-12.

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