
You use substitution to solve a system of equations and, after simplifying the equations, end with an expression that says 0 = 4. What does that say about the the lines and about the number of solutions to the system?


When you end up with an expression like \( 0 = 4 \) after simplifying a system of equations, this indicates that there is a **contradiction**. Such an expression cannot be true, because 0 is never equal to 4.

### What this means for the lines:
- **The lines are parallel.** When two lines are parallel, they never intersect, which means they have no point in common.
- This implies that the system of equations has **no solution**.

In other words, the system of equations is **inconsistent**. The equations represent two lines that do not intersect.

### Key Points:
- The contradiction \( 0 = 4 \) suggests the system is inconsistent.
- The lines represented by the system are parallel and therefore do not intersect.
- The system has **no solution**.

Would you like more details on how to identify inconsistent systems or how parallel lines behave algebraically? 

Here are 5 related questions:
1. How do you recognize when two lines are parallel from their equations?
2. What are the conditions under which two lines are parallel in a system of linear equations?
3. How do you interpret \( 0 = 0 \) in a system of equations?
4. What are the differences between inconsistent and dependent systems?
5. How do you graphically solve a system of equations?

**Tip**: When solving systems using substitution or elimination, if you reach a contradiction like \( 0 = 4 \), it’s an immediate indicator that the system has no solution and the lines are parallel.

You use substitution to solve a system of equations and, after simplifying the equations, end with an expression that says 0 = 4. What does that say about the lines and about the number of solutions to the system?

Explore why a system of equations leading to 0 = 4 means the lines are parallel and the system has no solutions. This problem demonstrates the concept of inconsistency in algebra and explains how to recognize when two lines have no intersection. Ideal for Grades 8-10 students.

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