https://file.aiquickdraw.com/mathbot/file/17287617898350o6yxdtg.jpg

The image shows a system of two linear equations that needs to be solved. The equations are:

\[
y = -\frac{10}{3}x - 25
\]
\[
y = -\frac{5}{2}x + 10
\]

We are tasked with determining whether this system has:

- **One solution**
- **No solution**
- **Infinite number of solutions**

### Solution Process:

Since both equations are in slope-intercept form \(y = mx + b\), we can approach this by setting the two expressions for \(y\) equal to each other and solving for \(x\):

\[
-\frac{10}{3}x - 25 = -\frac{5}{2}x + 10
\]

To eliminate the fractions, we can multiply through by 6 (the least common denominator of 3 and 2):

\[
6 \left(-\frac{10}{3}x - 25\right) = 6 \left(-\frac{5}{2}x + 10\right)
\]

Simplifying both sides:

\[
-20x - 150 = -15x + 60
\]

Now, isolate \(x\):

\[
-20x + 15x = 60 + 150
\]

\[
-5x = 210
\]

Solving for \(x\):

\[
x = -42
\]

Now that we have \(x = -42\), substitute this back into either of the original equations to find \(y\). Using the first equation:

\[
y = -\frac{10}{3}(-42) - 25
\]

Simplify:

\[
y = \frac{420}{3} - 25 = 140 - 25 = 115
\]

Thus, the solution to the system is \(x = -42\) and \(y = 115\), meaning **there is one unique solution**.

### Answer:
The correct option is **One solution**.

---

Would you like more details on any part of the process? Here are five related questions to explore further:

1. What does it mean when a system of equations has no solution?
2. How can you recognize when a system has infinitely many solutions?
3. What is the geometric interpretation of the solution to a system of two linear equations?
4. How can substitution be used to solve a system of equations like this?
5. What are other methods to solve systems of linear equations?

**Tip:** Always check your solution by substituting both values of \(x\) and \(y\) back into the original equations to ensure they hold true.

Solve the following system of equations: y = (-10/3)x - 25 and y = (-5/2)x + 10. Determine whether the system has one solution, no solution, or infinite solutions.

Learn how to solve a system of linear equations using substitution. This problem covers key algebraic concepts, including solving for x and y by equating the two equations and finding the unique solution for the system. Suitable for high school students.

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