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Gráfica lo en un plano cartesiano cada uno de los ejercicios que se encuentren con su respectivo solución y respuesta correctamente bien explicado

To address the provided exercises, we will solve the systems of linear equations step by step and match the solutions to the correct multiple-choice answers.

Let’s begin with problem 11:

### Problem 11:
System of equations:
\[
\begin{aligned}
x + y &= 12 \\
x - y &= 10
\end{aligned}
\]

**Step 1**: Add both equations to eliminate \( y \).
\[
(x + y) + (x - y) = 12 + 10
\]
\[
2x = 22 \quad \Rightarrow \quad x = 11
\]

**Step 2**: Substitute \( x = 11 \) into the first equation:
\[
11 + y = 12 \quad \Rightarrow \quad y = 1
\]

**Solution**: \( x = 11 \), \( y = 1 \). The solution set is \( (11, 1) \).

**Correct Answer**: a) CS = [11 ; 1].

---

### Problem 12:
System of equations:
\[
\begin{aligned}
x + 3y &= 1 \\
x - y &= 9
\end{aligned}
\]

**Step 1**: Solve the second equation for \( x \):
\[
x = y + 9
\]

**Step 2**: Substitute \( x = y + 9 \) into the first equation:
\[
(y + 9) + 3y = 1 \quad \Rightarrow \quad 4y + 9 = 1 \quad \Rightarrow \quad 4y = -8 \quad \Rightarrow \quad y = -2
\]

**Step 3**: Substitute \( y = -2 \) back into \( x = y + 9 \):
\[
x = -2 + 9 = 7
\]

**Solution**: \( x = 7 \), \( y = -2 \). The solution set is \( (7, -2) \).

**Correct Answer**: b) CS = [7 ; -2].

---

### Problem 13:
System of equations:
\[
\begin{aligned}
x + 3y &= 4 \\
x + 2y &= 3
\end{aligned}
\]

**Step 1**: Subtract the second equation from the first:
\[
(x + 3y) - (x + 2y) = 4 - 3 \quad \Rightarrow \quad y = 1
\]

**Step 2**: Substitute \( y = 1 \) into the first equation:
\[
x + 3(1) = 4 \quad \Rightarrow \quad x + 3 = 4 \quad \Rightarrow \quad x = 1
\]

**Solution**: \( x = 1 \), \( y = 1 \). The solution set is \( (1, 1) \).

**Correct Answer**: c) CS = [1 ; 1].

---

### Problem 14:
System of equations:
\[
\begin{aligned}
3x + 7y &= 23 \\
5x - 3y &= 9
\end{aligned}
\]

**Step 1**: Multiply the first equation by 3 and the second by 7 to eliminate \( y \):
\[
9x + 21y = 69 \\
35x - 21y = 63
\]

**Step 2**: Add the equations:
\[
9x + 35x = 69 + 63 \quad \Rightarrow \quad 44x = 132 \quad \Rightarrow \quad x = 3
\]

**Step 3**: Substitute \( x = 3 \) into the first equation:
\[
3(3) + 7y = 23 \quad \Rightarrow \quad 9 + 7y = 23 \quad \Rightarrow \quad 7y = 14 \quad \Rightarrow \quad y = 2
\]

**Solution**: \( x = 3 \), \( y = 2 \). The solution set is \( (3, 2) \).

**Correct Answer**: c) CS = [3 ; 2].

---

### Problem 15:
System of equations:
\[
\begin{aligned}
x + y &= 6 \\
2x + y &= 8
\end{aligned}
\]

**Step 1**: Subtract the first equation from the second:
\[
(2x + y) - (x + y) = 8 - 6 \quad \Rightarrow \quad x = 2
\]

**Step 2**: Substitute \( x = 2 \) into the first equation:
\[
2 + y = 6 \quad \Rightarrow \quad y = 4
\]

**Solution**: \( x = 2 \), \( y = 4 \). The solution set is \( (2, 4) \).

**Correct Answer**: a) CS = [2 ; 4].

---

Would you like to continue with the rest of the problems or need additional explanations? Feel free to ask for further clarification!

---

Here are five follow-up questions:
1. How do you graphically represent the solutions of two linear equations?
2. Why is it useful to add or subtract equations in systems of equations?
3. What happens if the system of equations has no solution or infinitely many solutions?
4. Can we always use substitution to solve systems of equations, and why?
5. How does the coefficient method help in solving systems of equations?

**Tip**: When solving a system of equations, always double-check by substituting the found values of \( x \) and \( y \) back into both original equations to ensure correctness.

Gráfica lo en un plano cartesiano cada uno de los ejercicios que se encuentren con su respectivo solución y respuesta correctamente bien explicado.

Learn how to solve systems of linear equations step by step with detailed explanations and graphical representation. These problems cover algebraic concepts, including the elimination and substitution methods. Ideal for students in Grades 8-10.

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