Use the substitution method or the elimination method to solve the system.
x
minus
3 y
equals
16
4 x
minus
12 y
equals
16
 


We are given the system of equations:

1. \( x - 3y = 16 \)
2. \( 4x - 12y = 16 \)

Let's solve this using both the **substitution method** and the **elimination method** for clarity.

### Substitution Method:

1. From the first equation, solve for \( x \) in terms of \( y \):
   \[
   x = 16 + 3y
   \]

2. Substitute this expression for \( x \) into the second equation:
   \[
   4(16 + 3y) - 12y = 16
   \]

3. Expand and simplify:
   \[
   64 + 12y - 12y = 16
   \]
   \[
   64 = 16
   \]

Since this leads to a contradiction, the system has **no solution**.

### Elimination Method:

1. Multiply the first equation by 4 to align it with the second equation:
   \[
   4(x - 3y) = 4(16)
   \]
   \[
   4x - 12y = 64
   \]

2. Now, subtract the second equation from this:
   \[
   (4x - 12y) - (4x - 12y) = 64 - 16
   \]
   \[
   0 = 48
   \]

This is another contradiction, confirming that the system has **no solution**.

### Conclusion:
The system is inconsistent and has no solution.

---

Would you like more details on how this works? Here are five related questions to explore further:

1. What does it mean when a system of equations has no solution?
2. How can we recognize inconsistent systems visually through graphing?
3. What are the differences between substitution and elimination methods?
4. Can a system of equations have exactly one solution? If so, how do we identify it?
5. What is a dependent system of equations, and how is it solved?

**Tip:** When solving a system of equations, always check for consistency. If you encounter a contradiction (like \( 0 = 48 \)), the system has no solution.

Use the substitution method or the elimination method to solve the system. x - 3y = 16 and 4x - 12y = 16.

Learn how to solve the system of linear equations x - 3y = 16 and 4x - 12y = 16 using both substitution and elimination methods. Discover how to identify an inconsistent system, leading to no solution, and understand the mathematical process behind this outcome. Suitable for students in Grades 8-10.

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