We are given two equations:

1. $a^2 + b^2 = 25$
2. $a + b = 7$

To solve for $a$ and $b$, we can use substitution or elimination. Here's how we can approach it:

Step 1: Express $a^2 + b^2$ in terms of $(a+b)^2$

We know that:

$(a+b)^2 = a^2 + 2ab + b^2$

Using the second equation $a + b = 7$:

$7^2 = a^2 + 2ab + b^2$ $49 = a^2 + 2ab + b^2$

Now substitute $a^2 + b^2 = 25$ from the first equation into this expression:

$49 = 25 + 2ab$

Step 2: Solve for $ab$

Subtract 25 from both sides:

$24 = 2ab$

Divide by 2:

$ab = 12$

Step 3: Solve the quadratic equation

Now that we have both $a + b = 7$ and $ab = 12$, we can treat $a$ and $b$ as the roots of the quadratic equation:

$x^2 - (a+b)x + ab = 0$

Substitute $a+b = 7$ and $ab = 12$:

$x^2 - 7x + 12 = 0$

Step 4: Factor the quadratic equation

Factor the quadratic:

$(x - 3)(x - 4) = 0$

So, $x = 3$ or $x = 4$.

Step 5: Conclusion

Therefore, the values of $a$ and $b$ are $a = 3$, $b = 4$ (or vice versa: $a = 4$, $b = 3$).

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Would you like more details on any part of this solution?

Here are 5 related questions:

1. How do you solve quadratic equations by factoring?
2. Can we solve this system using substitution instead of factoring?
3. What happens if we change the constants in the equations, such as $a^2 + b^2 = 36$?
4. How do we solve a system of nonlinear equations graphically?
5. How can the quadratic formula be applied to this system?

Tip: Always check your solution by substituting the values of variables back into the original equations.