You have two equations here:

1. $y = x + 7$
2. $yx = -12$

We can solve this system of equations by substituting the expression for $y$ from the first equation into the second equation.

Step 1: Substitute $y = x + 7$ into $yx = -12$

Since $y = x + 7$, replace $y$ in the second equation:

$(x + 7)x = -12$

This simplifies to:

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

Step 2: Solve the quadratic equation

Now we have a quadratic equation:

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

To solve this, we can factor the equation. We need two numbers that multiply to $12$ and add up to $7$. Those numbers are $3$ and $4$.

So, we can factor as:

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

Step 3: Find the values of $x$

Set each factor equal to zero:

1. $x + 3 = 0 \Rightarrow x = -3$
2. $x + 4 = 0 \Rightarrow x = -4$

Step 4: Find the corresponding values of $y$

Now, substitute these values of $x$ back into $y = x + 7$ to find $y$:

• For $x = -3$: $y = -3 + 7 = 4$

• For $x = -4$: $y = -4 + 7 = 3$

Solution

The solutions to the system are:

1. $(x, y) = (-3, 4)$
2. $(x, y) = (-4, 3)$

Would you like a deeper explanation on any part of this process?

Here are some related questions you may find helpful:

1. How do you solve systems of equations by substitution?
2. What are the steps to factor quadratic equations?
3. How can we confirm the solutions to a system of nonlinear equations?
4. What other methods exist to solve systems of equations?
5. How do substitution and elimination methods compare in solving systems of equations?

Tip: Always check your solutions by substituting them back into the original equations to ensure accuracy.