5x-y=9 2x+y=5 Which of the following points is a solution to the following system of equations? (2,1)

(1,−4)

(3,−1)

(−1,7)

To determine which of the given points is a solution to the system of equations:

\[
\begin{aligned}
5x - y &= 9 \quad \text{(Equation 1)} \\
2x + y &= 5 \quad \text{(Equation 2)}
\end{aligned}
\]

we'll substitute each point into both equations to check if it satisfies both.

1. **Point (2,1):**
   - Substitute \( x = 2 \) and \( y = 1 \) into Equation 1:
     \[
     5(2) - 1 = 10 - 1 = 9 \quad (\text{True})
     \]
   - Substitute \( x = 2 \) and \( y = 1 \) into Equation 2:
     \[
     2(2) + 1 = 4 + 1 = 5 \quad (\text{True})
     \]
   - **(2,1) is a solution.**

2. **Point (1,−4):**
   - Substitute \( x = 1 \) and \( y = -4 \) into Equation 1:
     \[
     5(1) - (-4) = 5 + 4 = 9 \quad (\text{True})
     \]
   - Substitute \( x = 1 \) and \( y = -4 \) into Equation 2:
     \[
     2(1) + (-4) = 2 - 4 = -2 \quad (\text{False})
     \]
   - **(1,−4) is not a solution.**

3. **Point (3,−1):**
   - Substitute \( x = 3 \) and \( y = -1 \) into Equation 1:
     \[
     5(3) - (-1) = 15 + 1 = 16 \quad (\text{False})
     \]
   - **(3,−1) is not a solution.**

4. **Point (−1,7):**
   - Substitute \( x = -1 \) and \( y = 7 \) into Equation 1:
     \[
     5(-1) - 7 = -5 - 7 = -12 \quad (\text{False})
     \]
   - **(−1,7) is not a solution.**

**Conclusion:** The point \((2,1)\) is the only solution to the system of equations.

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you solve a system of linear equations using the substitution method?
2. How can you graphically represent the solution to a system of linear equations?
3. What are the different methods to solve a system of linear equations?
4. How can you determine if a system of equations has no solution or infinitely many solutions?
5. What is the importance of checking both equations when verifying a solution to a system?

**Tip:** Always verify the solution by substituting back into both equations to ensure accuracy.

Learn how to identify solutions to the system of linear equations 5x - y = 9 and 2x + y = 5. This problem involves applying the substitution method to verify which points satisfy both equations. Suitable for students in Grades 7-9.

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