https://file.aiquickdraw.com/mathbot/file/1728001308536wb7pzqtm.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
5x + 6y &= 0 \\
-9x + 7y &= 0
\end{aligned}
\]

Let's solve this system step by step.

### Step 1: Eliminate one variable
We can use the method of elimination to eliminate one of the variables. Let's multiply the first equation by 9 and the second equation by 5 so that the coefficients of \(x\) are opposites:

\[
9(5x + 6y) = 9 \cdot 0 \quad \text{and} \quad 5(-9x + 7y) = 5 \cdot 0
\]

This simplifies to:

\[
45x + 54y = 0
\]
\[
-45x + 35y = 0
\]

### Step 2: Add the two equations
Now, add the two equations to eliminate \(x\):

\[
(45x + 54y) + (-45x + 35y) = 0 + 0
\]

This simplifies to:

\[
89y = 0
\]

### Step 3: Solve for \(y\)
\[
y = 0
\]

### Step 4: Solve for \(x\)
Now that we know \(y = 0\), substitute this value into one of the original equations. Let's use the first equation:

\[
5x + 6(0) = 0
\]

This simplifies to:

\[
5x = 0
\]

\[
x = 0
\]

### Final Solution:
The system has a unique solution:

\[
x = 0, \quad y = 0
\]

There is only one solution, and the coordinates are \((0, 0)\).

Would you like more details on any step, or do you have any questions? 

Here are 5 related questions to further your understanding:
1. What other methods can be used to solve this system of equations?
2. How would the solution change if there were no solutions?
3. Can this method be applied to nonlinear systems?
4. What does it mean if a system of equations has infinitely many solutions?
5. How can you verify the solution to a system of equations?

**Tip:** Always double-check your solution by substituting the values of \(x\) and \(y\) back into both original equations!

Solve the following system of linear equations: 5x + 6y = 0, -9x + 7y = 0. If there is one solution, give its coordinates. If the system has infinitely many solutions, enter 'z' in the answer blank for x and enter a formula for y in terms of z. If no solutions, enter 'none'.

Learn how to solve the linear system of equations 5x + 6y = 0 and -9x + 7y = 0 step by step using the elimination method. This problem covers essential algebraic concepts suitable for students in Grades 10-12.

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