To solve this system of simultaneous equations:

1. Equations Given: $2x + y = 5$ $x^2 + y^2 = 25$

2. Step 1: Express $y$ in terms of $x$: From the first equation, isolate $y$: $y = 5 - 2x$

3. Step 2: Substitute $y = 5 - 2x$ into the second equation: Substitute $y = 5 - 2x$ into $x^2 + y^2 = 25$: $x^2 + (5 - 2x)^2 = 25$

4. Step 3: Expand and Simplify: Expanding $(5 - 2x)^2$: $x^2 + (25 - 20x + 4x^2) = 25$ Simplify this expression: $5x^2 - 20x + 25 = 25$ Subtract 25 from both sides: $5x^2 - 20x = 0$ Factor out $5x$: $5x(x - 4) = 0$ So, $x = 0$ or $x = 4$.

5. Step 4: Find corresponding values of $y$:

   • If $x = 0$: $y = 5 - 2(0) = 5$ So, one solution is $(x, y) = (0, 5)$.
   • If $x = 4$: $y = 5 - 2(4) = 5 - 8 = -3$ So, the other solution is $(x, y) = (4, -3)$.

6. Solutions: The solutions to the system are: $(x, y) = (0, 5) \quad \text{and} \quad (x, y) = (4, -3)$

Would you like a further breakdown of these steps, or do you have any questions about this solution?

Here are some additional questions to deepen your understanding:

1. How would you verify that these solutions satisfy both original equations?
2. Could there be any other solutions to these equations?
3. What geometric shapes do these equations represent, and where do they intersect?
4. How might you solve this system using a different method, such as graphing?
5. What happens if you change one of the constants in the equations?

Tip: When solving systems of equations, substituting one variable in terms of another can simplify solving for both variables.