Let's solve the system of equations and analyze its solutions.

The given system is:

1. $y = -4x + 9y$
2. $y = \frac{2}{3}x - 5$

Step 1: Simplify the first equation

Start by simplifying the first equation:
$y = -4x + 9y$ Subtract $9y$ from both sides: $y - 9y = -4x$ $-8y = -4x$ Now, divide both sides by $-8$: $y = \frac{1}{2}x$

So the first equation simplifies to: $y = \frac{1}{2}x$

Step 2: Set the two equations equal

Now that we have the simplified form of the first equation $y = \frac{1}{2}x$ and the second equation $y = \frac{2}{3}x - 5$, we can set them equal to each other to solve for $x$.

$\frac{1}{2}x = \frac{2}{3}x - 5$

Step 3: Solve for $x$

To eliminate the fractions, multiply through by 6 (the least common denominator of 2 and 3): $6 \times \frac{1}{2}x = 6 \times \left( \frac{2}{3}x - 5 \right)$ This simplifies to: $3x = 4x - 30$ Now, subtract $4x$ from both sides: $3x - 4x = -30$ $-x = -30$ Multiply both sides by $-1$: $x = 30$

Step 4: Solve for $y$

Now substitute $x = 30$ into either of the original equations to solve for $y$. Let's use the second equation: $y = \frac{2}{3}x - 5$ Substitute $x = 30$: $y = \frac{2}{3}(30) - 5$ $y = 20 - 5 = 15$

Step 5: Conclusion

The solution to the system of equations is $x = 30$ and $y = 15$, which means the system has one solution.

Final Answer:

The system of equations has one solution, which is $x = 30$ and $y = 15$.

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Would you like further explanation on any step, or do you have any additional questions?

Here are some related questions you can explore:

1. What happens if the coefficients of $x$ in both equations are the same?
2. How would you handle a system of equations with no solution?
3. What if the equations were both in standard form, $Ax + By = C$?
4. How do you solve a system of equations using substitution?
5. How would the solution change if one of the equations was nonlinear?

Tip: When dealing with a system of linear equations, always simplify the equations first and check for possible points of intersection graphically for a better understanding of the solution!