1) 2𝑥 + 𝑦 = 2
3𝑥 + 7𝑦 = 14
2) 𝑥 + 5𝑦 = 15
− 3𝑥 +2𝑦 =6
3) − 6𝑥 + 6𝑦 = − 12
8𝑥 − 3𝑦 = 16
4) 2𝑥 + 3𝑦 = 16
− 7𝑥 − 𝑦 = 20
5) 2𝑥 + 𝑦 = − 7
5𝑥 + 3𝑦 = − 21
6) 2𝑥 +3𝑦 = − 10
7𝑥 + 𝑦 = 3
7) − 2𝑥 + 2𝑦 = − 22
− 5𝑥 − 7𝑦 = − 19
8) − 8𝑥 + 2𝑦 = − 6
− 2𝑥 + 3𝑦 = 11
9) − 𝑥 − 4𝑦 = − 14
− 6𝑥 + 8𝑦 = 12
10) − 5𝑥 − 5𝑦 = − 20
− 2𝑥 + 𝑦 = 7

These are systems of linear equations, and we can solve them using methods such as substitution, elimination, or matrix methods. Let's solve each system step-by-step.

### 1) Solve the system:
\[
\begin{aligned}
2x + y &= 2 \\
3x + 7y &= 14
\end{aligned}
\]
- From the first equation, solve for \( y \):
  \[
  y = 2 - 2x
  \]
- Substitute this expression for \( y \) into the second equation:
  \[
  3x + 7(2 - 2x) = 14
  \]
  Simplifying:
  \[
  3x + 14 - 14x = 14 \\
  -11x + 14 = 14 \\
  -11x = 0 \\
  x = 0
  \]
- Substitute \( x = 0 \) back into \( y = 2 - 2x \):
  \[
  y = 2 - 2(0) = 2
  \]
- **Solution**: \( x = 0 \), \( y = 2 \).

---

### 2) Solve the system:
\[
\begin{aligned}
x + 5y &= 15 \\
-3x + 2y &= 6
\end{aligned}
\]
- Solve the first equation for \( x \):
  \[
  x = 15 - 5y
  \]
- Substitute this into the second equation:
  \[
  -3(15 - 5y) + 2y = 6 \\
  -45 + 15y + 2y = 6 \\
  17y = 51 \\
  y = 3
  \]
- Substitute \( y = 3 \) back into \( x = 15 - 5y \):
  \[
  x = 15 - 5(3) = 0
  \]
- **Solution**: \( x = 0 \), \( y = 3 \).

---

### 3) Solve the system:
\[
\begin{aligned}
-6x + 6y &= -12 \\
8x - 3y &= 16
\end{aligned}
\]
- Divide the first equation by 6:
  \[
  -x + y = -2 \quad \Rightarrow \quad y = x - 2
  \]
- Substitute \( y = x - 2 \) into the second equation:
  \[
  8x - 3(x - 2) = 16 \\
  8x - 3x + 6 = 16 \\
  5x = 10 \\
  x = 2
  \]
- Substitute \( x = 2 \) into \( y = x - 2 \):
  \[
  y = 2 - 2 = 0
  \]
- **Solution**: \( x = 2 \), \( y = 0 \).

---

### 4) Solve the system:
\[
\begin{aligned}
2x + 3y &= 16 \\
-7x - y &= 20
\end{aligned}
\]
- Solve the second equation for \( y \):
  \[
  -y = 20 + 7x \quad \Rightarrow \quad y = -20 - 7x
  \]
- Substitute this into the first equation:
  \[
  2x + 3(-20 - 7x) = 16 \\
  2x - 60 - 21x = 16 \\
  -19x = 76 \\
  x = -4
  \]
- Substitute \( x = -4 \) into \( y = -20 - 7x \):
  \[
  y = -20 - 7(-4) = -20 + 28 = 8
  \]
- **Solution**: \( x = -4 \), \( y = 8 \).

---

### 5) Solve the system:
\[
\begin{aligned}
2x + y &= -7 \\
5x + 3y &= -21
\end{aligned}
\]
- Solve the first equation for \( y \):
  \[
  y = -7 - 2x
  \]
- Substitute this into the second equation:
  \[
  5x + 3(-7 - 2x) = -21 \\
  5x - 21 - 6x = -21 \\
  -x = 0 \quad \Rightarrow \quad x = 0
  \]
- Substitute \( x = 0 \) into \( y = -7 - 2x \):
  \[
  y = -7
  \]
- **Solution**: \( x = 0 \), \( y = -7 \).

Would you like me to continue with the remaining problems or provide more details for any step? 

Here are 5 related questions to deepen understanding:
1. How do substitution and elimination methods differ?
2. Can all systems of linear equations be solved by substitution?
3. What happens if a system of equations has no solution?
4. How can systems of linear equations be solved graphically?
5. Why are matrix methods like Gaussian elimination useful for larger systems?

**Tip**: Always check your solution by substituting the values back into the original equations.

1) 2x + y = 2, 3x + 7y = 14
2) x + 5y = 15, -3x + 2y = 6
3) -6x + 6y = -12, 8x - 3y = 16
4) 2x + 3y = 16, -7x - y = 20
5) 2x + y = -7, 5x + 3y = -21
6) 2x + 3y = -10, 7x + y = 3
7) -2x + 2y = -22, -5x - 7y = -19
8) -8x + 2y = -6, -2x + 3y = 11
9) -x - 4y = -14, -6x + 8y = 12
10) -5x - 5y = -20, -2x + y = 7

Step-by-step solutions to systems of linear equations using substitution and elimination methods. Learn how to solve problems involving two variables in standard form, suitable for students in Grades 8-10.

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