Solve for x and y.{𝑥+𝑦=7𝑥𝑦=12

To solve the system of equations:
\[ \begin{cases} 
x + y = 7 \\
xy = 12 
\end{cases} \]

Let's use substitution and quadratic methods.

First, express \( y \) in terms of \( x \) from the first equation:
\[ y = 7 - x \]

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

This simplifies to:
\[ 7x - x^2 = 12 \]

Rearrange to form a standard quadratic equation:
\[ x^2 - 7x + 12 = 0 \]

Next, solve the quadratic equation:
\[ x^2 - 7x + 12 = 0 \]

We can solve this by factoring. We look for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. So, we can factor the quadratic as:
\[ (x - 3)(x - 4) = 0 \]

Set each factor to zero and solve for \( x \):
\[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \]
\[ x = 3 \quad \text{or} \quad x = 4 \]

Substitute these values back into the equation \( y = 7 - x \) to find the corresponding \( y \) values:
- If \( x = 3 \):
  \[ y = 7 - 3 = 4 \]

- If \( x = 4 \):
  \[ y = 7 - 4 = 3 \]

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

Would you like more details or have any questions? Here are 5 more questions you might be interested in:

1. How do you solve a system of linear equations using the substitution method?
2. What is the quadratic formula and how is it used?
3. Can you explain the factoring method for solving quadratic equations?
4. How do you solve systems of equations using the elimination method?
5. What are the properties of polynomial equations?

**Tip:** When solving quadratic equations, always try factoring first as it is often the simplest method. If factoring is difficult, use the quadratic formula.

Learn how to solve the system of equations x + y = 7 and xy = 12 using the substitution and quadratic methods. Step-by-step solution with detailed explanations and examples. Suitable for high school students and anyone preparing for algebra exams.

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