Solve for x and y.{𝑥+𝑦=7𝑥𝑦=12

To solve for \(x\) and \(y\) given the system of equations:

\[
\begin{cases}
x + y = 7 \\
xy = 12
\end{cases}
\]

we can follow these steps:

1. From the first equation, express \(y\) in terms of \(x\):
   \[
   y = 7 - x
   \]

2. Substitute \(y = 7 - x\) into the second equation:
   \[
   x(7 - x) = 12
   \]

3. Simplify and rearrange the equation to form a quadratic equation:
   \[
   7x - x^2 = 12 \implies x^2 - 7x + 12 = 0
   \]

4. Solve the quadratic equation \(x^2 - 7x + 12 = 0\). We can factorize it as:
   \[
   (x - 3)(x - 4) = 0
   \]

5. Set each factor to zero to find the solutions for \(x\):
   \[
   x - 3 = 0 \quad \Rightarrow \quad x = 3
   \]
   \[
   x - 4 = 0 \quad \Rightarrow \quad x = 4
   \]

6. Use \(y = 7 - x\) to find the corresponding values of \(y\) for each \(x\):
   - For \(x = 3\):
     \[
     y = 7 - 3 = 4
     \]
   - For \(x = 4\):
     \[
     y = 7 - 4 = 3
     \]

Thus, the solutions to the system are:
\[
(x, y) = (3, 4) \quad \text{and} \quad (4, 3)
\]

Do you want details or have any questions?

Here are some more questions you might want to ask:
1. How do you solve a quadratic equation using the quadratic formula?
2. Can you explain how to factorize a quadratic equation?
3. What are some methods to solve systems of linear equations?
4. How do you graph the solutions of a system of equations?
5. What are some applications of solving systems of equations in real life?

**Tip:** When solving systems of equations, substituting one equation into another can often simplify the problem to a single-variable equation.

Learn how to solve the system of equations x + y = 7 and xy = 12 step-by-step with detailed solutions. This problem covers systems of linear equations and quadratic equations, including the use of substitution and factoring methods. Suitable for students in Grades 9-12.

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