To solve the given system of equations using the addition method (also known as the elimination method), follow these steps:

System of Equations:

3x + 2y = 0 \tag{1} 9x + 6y = 8 \tag{2}

Step 1: Make the coefficients of $x$ or $y$ equal.

We can multiply equation (1) by $-3$ to match the coefficients of $x$ in both equations.

$-3(3x + 2y) = -3(0)$ -9x - 6y = 0 \tag{3}

Step 2: Add the new equation to equation (2).

Now add equation (3) to equation (2):

$(-9x - 6y) + (9x + 6y) = 0 + 8$ $0 = 8$

Step 3: Analyze the result.

The result, $0 = 8$, is a contradiction. This means that the given system of equations is inconsistent and has no solution. The lines represented by these equations are parallel, so they do not intersect.

Final Answer:

The system of equations has no solution.

Would you like more details on this solution or have any questions?

Related Questions:

1. How do you determine if a system of equations has no solution?
2. What does it mean for two lines to be parallel in terms of their slopes?
3. Can the addition method be used for non-linear systems?
4. How can you verify a solution for a system of equations graphically?
5. What is the difference between inconsistent, consistent, and dependent systems?

Tip:

When using the addition method, if the resulting equation is a contradiction (like $0 = 8$), the system has no solution. If you get a true statement (like $0 = 0$), the system may have infinitely many solutions.