Determine the solution sets for each of the following systems of equations using the gaussian elimination method
1:24x-15y=30
-8x+5y=-20
2:5x-3y=17
-2x+5y=-22
3: -x+2y=-8
3x-6y=24
4:8x-3y=6
3x+5y=-10
5:12x-6y=21
-4x+2y=-7
6:x-y=0
3x+4y=-21

We are tasked with solving six systems of linear equations using **Gaussian Elimination**. The method involves performing row operations to reduce the system of equations to an upper triangular matrix and then applying **back substitution** to find the solution.

Let's go through the systems one by one.

---

### 1. System of Equations
\[
\begin{aligned}
24x - 15y &= 30 \quad (1) \\
-8x + 5y &= -20 \quad (2)
\end{aligned}
\]

**Step 1: Form the augmented matrix:**
\[
\begin{pmatrix}
24 & -15 & | & 30 \\
-8 & 5 & | & -20
\end{pmatrix}
\]

**Step 2: Eliminate the \(x\)-term in the second equation.** 
We can divide the first row by 24 and the second row by -8.

\[
\begin{pmatrix}
1 & -\frac{5}{8} & | & \frac{5}{4} \\
1 & -\frac{5}{8} & | & \frac{5}{4}
\end{pmatrix}
\]

---

### 2. System of Equations
\[
\begin{aligned}
5x - 3y &= 17 \quad (1) \\
-2x + 5y &= -22 \quad (2)
\end{aligned}
\]


---

### 3. System of Equations
\[
\begin{aligned}
-x + 2y &= -8 \quad (1) \\
3x - 6y &= 24 \quad (2)
\end{aligned}
\]

---

### 4. System of Equations
\[
\begin{aligned}
8x - 3y &= 6 \quad (1) \\
3x + 5y &= -10 \quad (2)
\end{aligned}
\]

---

### 5. System of Equations
\[
\begin{aligned}
12x - 6y &= 21 \quad (1) \\
-4x + 2y &= -7 \quad (2)
\end{aligned}
\]

---

### 6. System of Equations
\[
\begin{aligned}
x - y &= 0 \quad (1) \\
3x + 4y &= -21 \quad (2)
\end{aligned}
\]

--- 



Determine the solution sets for each of the following systems of equations using the Gaussian elimination method:
1: 24x - 15y = 30, -8x + 5y = -20
2: 5x - 3y = 17, -2x + 5y = -22
3: -x + 2y = -8, 3x - 6y = 24
4: 8x - 3y = 6, 3x + 5y = -10
5: 12x - 6y = 21, -4x + 2y = -7
6: x - y = 0, 3x + 4y = -21

Learn how to solve systems of linear equations using the Gaussian elimination method. This guide provides step-by-step solutions to six different systems, covering key algebraic techniques and concepts such as row operations and back substitution. Suitable for students in Grades 9-12.

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