Solve the system.

−6x+2y=14-6x+2y=14

−9x+3y=23

You are given the system of two linear equations:

1. \(-6x + 2y = 14\)
2. \(-9x + 3y = 23\)

Let's solve this system step by step.

### Step 1: Simplify the equations (if needed).
Both equations are already simplified, so we proceed with solving them.

### Step 2: Use elimination or substitution.
We can use the **elimination method** by multiplying both equations to make the coefficients of one variable equal, then subtracting them to eliminate that variable.

First, let's make the coefficients of \( y \) equal. To do this, we need to multiply both equations so that the \( y \)-coefficients are the same.

- Multiply the first equation by 3:
  \[
  3(-6x + 2y) = 3(14) \implies -18x + 6y = 42
  \]
- Multiply the second equation by 2:
  \[
  2(-9x + 3y) = 2(23) \implies -18x + 6y = 46
  \]

### Step 3: Eliminate a variable.
Now subtract the first modified equation from the second modified equation:

\[
(-18x + 6y) - (-18x + 6y) = 46 - 42
\]

This simplifies to:

\[
0 = 4
\]

### Step 4: Interpret the result.
The equation \( 0 = 4 \) is a contradiction, meaning that there is no solution to this system of equations. Therefore, the system is **inconsistent**.

### Conclusion:
The system of equations has **no solution**.

---

Do you have any questions or need further details? Here are some related questions to think about:

1. What does it mean for a system to be inconsistent?
2. How can you graphically interpret inconsistent systems of equations?
3. What are the other methods to solve systems of equations (e.g., substitution)?
4. What is the difference between a dependent and independent system of equations?
5. How do you classify systems as having one solution, no solution, or infinitely many solutions?

**Tip:** If a system of linear equations leads to a contradiction (like \(0 = 4\)), the system has no solution and is classified as inconsistent.

Solve the system. −6x + 2y = 14 and −9x + 3y = 23

Explore how to solve the system of linear equations −6x + 2y = 14 and −9x + 3y = 23 using the elimination method. This problem leads to an inconsistent system, showing no solution. Detailed steps and mathematical principles are provided for students in Grades 9-11.

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