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Let's work through the problems step by step.

### Problem 1:
**Determine whether the ordered pair is a solution to the system.**

1. \((3, 2)\)  
   \[
   4x - 3y = 6 \\
   x - 4 = y
   \]

**Solution:**

1. **Substitute the ordered pair \((3, 2)\) into the first equation:**
   \[
   4(3) - 3(2) = 12 - 6 = 6
   \]
   This is true, so \((3, 2)\) satisfies the first equation.

2. **Substitute into the second equation:**
   \[
   3 - 4 = -1 \quad \text{and} \quad y = 2
   \]
   This is false, so \((3, 2)\) does **not** satisfy the second equation.

Therefore, \((3, 2)\) is **not** a solution to the system.

### Problem 2:
**How many solutions does the system have?**

\[
4x + y = 1 \\
3y - 2x = 5
\]

**Solution:**

To find the number of solutions, let's first solve the system using substitution or elimination:

1. Solve the first equation for \(y\):
   \[
   y = 1 - 4x
   \]
2. Substitute into the second equation:
   \[
   3(1 - 4x) - 2x = 5 \\
   3 - 12x - 2x = 5 \\
   -14x = 2 \\
   x = -\frac{1}{7}
   \]
   Now, find \(y\):
   \[
   y = 1 - 4(-\frac{1}{7}) = 1 + \frac{4}{7} = \frac{11}{7}
   \]
So, the system has exactly **one solution**: \((-1/7, 11/7)\).

### Problem 3:
**How many solutions does the system have?**

\[
8y + 9 = 7x \\
24y + 27 = 21x
\]

**Solution:**

Simplify both equations by dividing the second by 3:
\[
8y + 9 = 7x \\
8y + 9 = 7x
\]

Both equations are identical, so there are **infinitely many solutions.**

### Problem 5:
**Solve the system using substitution or elimination.**

\[
-2x + y = 0 \\
-x + y = 2
\]

**Solution:**

1. Solve the first equation for \(y\):
   \[
   y = 2x
   \]
2. Substitute into the second equation:
   \[
   -x + 2x = 2 \\
   x = 2
   \]
   Substitute \(x = 2\) into the first equation to find \(y\):
   \[
   y = 4
   \]
   The solution is \((2, 4)\).

### Problem 6:
**Solve the system using substitution or elimination.**

\[
6x + 2y = 11 \\
y = -4x + 6
\]

**Solution:**

1. Substitute the second equation into the first:
   \[
   6x + 2(-4x + 6) = 11 \\
   6x - 8x + 12 = 11 \\
   -2x = -1 \\
   x = \frac{1}{2}
   \]
   Now, find \(y\):
   \[
   y = -4\left(\frac{1}{2}\right) + 6 = 4
   \]
   The solution is \(\left(\frac{1}{2}, 4\right)\).

Would you like further details or explanations on these? Also, here are five related questions:

1. How do you solve systems of linear equations graphically?
2. What does it mean for a system to have infinitely many solutions?
3. How can you check your solution to a system of equations?
4. What are some common mistakes to avoid when solving systems of equations?
5. How do you apply systems of equations to real-world problems?

**Tip:** Always double-check your substitutions and signs to avoid errors when solving equations!

Explore detailed solutions for solving systems of linear equations step-by-step. Learn about different methods such as substitution and elimination. Includes examples and explanations suitable for high school students.

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