https://file.aiquickdraw.com/mathbot/file/1730485284369expwdgak.jpg

 **A.** 

A

 **B.** 

B

 **C.** 

C

 **D.** 

D

To determine which set of coordinates satisfies the system of equations \( y = x - 3 \) and \( y = -2x + 1 \), we need to find the intersection point of these two lines. This point represents the solution to the system, where both equations are simultaneously true.

Let's solve for the intersection algebraically by setting the equations equal to each other:

1. **Equate the two equations**:
   \[
   x - 3 = -2x + 1
   \]

2. **Solve for \( x \)**:
   - Move \( -2x \) to the left side:
     \[
     x + 2x - 3 = 1
     \]
   - Simplify:
     \[
     3x - 3 = 1
     \]
   - Add 3 to both sides:
     \[
     3x = 4
     \]
   - Divide by 3:
     \[
     x = \frac{4}{3}
     \]

3. **Substitute \( x = \frac{4}{3} \) back into one of the equations to solve for \( y \)**:
   - Using \( y = x - 3 \):
     \[
     y = \frac{4}{3} - 3 = \frac{4}{3} - \frac{9}{3} = -\frac{5}{3}
     \]

So, the intersection point is \( \left( \frac{4}{3}, -\frac{5}{3} \right) \).

Now, let's compare this result with the points labeled in the graph:

- **A**: Does not appear at \( \left( \frac{4}{3}, -\frac{5}{3} \right) \).
- **B**: Does not match \( \left( \frac{4}{3}, -\frac{5}{3} \right) \).
- **C**: Does not match \( \left( \frac{4}{3}, -\frac{5}{3} \right) \).
- **D**: This point corresponds to \( \left( \frac{4}{3}, -\frac{5}{3} \right) \).

### Conclusion:
The correct answer is **D**. 

Would you like further explanation on any part of the solution?

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### Further Questions:
1. How can you verify graphically that a point lies on both lines of a system of equations?
2. How do you interpret the intersection of two lines in terms of solutions for a system of linear equations?
3. What happens if two lines are parallel in a system of equations?
4. How would you solve a system of linear equations using the substitution method?
5. What is the elimination method, and how does it work for solving systems?

#### Tip:
For checking solutions to systems of equations, always look for the point where the lines intersect if they do, as that represents the unique solution.

Which set of coordinates satisfies the system of equations y = x - 3 and y = -2x + 1?

Understand how to find the intersection point for the system of equations y = x - 3 and y = -2x + 1 by solving algebraically. This step-by-step solution explores key concepts in coordinate geometry and systems of equations, suitable for students in Grades 8-10.

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