https://file.aiquickdraw.com/mathbot/file/1726447006528a5rkwxul.jpg

The system of equations given in the image is:

\[
xy = 16
\]
\[
2x - y = -4
\]

We can solve this system step by step.

### Step 1: Solve for \( y \) in the second equation
The second equation is \( 2x - y = -4 \). We can solve for \( y \) in terms of \( x \):
\[
y = 2x + 4
\]

### Step 2: Substitute \( y = 2x + 4 \) into the first equation
Now substitute this expression for \( y \) into the first equation \( xy = 16 \):
\[
x(2x + 4) = 16
\]

### Step 3: Expand and solve the quadratic equation
Expanding the equation gives:
\[
2x^2 + 4x = 16
\]
Now, bring all terms to one side:
\[
2x^2 + 4x - 16 = 0
\]
Divide through by 2 to simplify:
\[
x^2 + 2x - 8 = 0
\]

### Step 4: Solve the quadratic equation
This quadratic equation can be solved using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 2 \), and \( c = -8 \). Plugging these values into the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)}
\]
\[
x = \frac{-2 \pm \sqrt{4 + 32}}{2}
\]
\[
x = \frac{-2 \pm \sqrt{36}}{2}
\]
\[
x = \frac{-2 \pm 6}{2}
\]
Thus, the two solutions for \( x \) are:
\[
x = \frac{-2 + 6}{2} = 2 \quad \text{or} \quad x = \frac{-2 - 6}{2} = -4
\]

### Step 5: Find corresponding \( y \) values
Now, substitute each \( x \) value into \( y = 2x + 4 \) to find the corresponding \( y \):

- When \( x = 2 \):
  \[
  y = 2(2) + 4 = 4 + 4 = 8
  \]
- When \( x = -4 \):
  \[
  y = 2(-4) + 4 = -8 + 4 = -4
  \]

### Final Solution
The solutions to the system are:
\[
(x, y) = (2, 8) \quad \text{or} \quad (x, y) = (-4, -4)
\]

Would you like further details or have any questions about this solution?

Here are 5 related questions for further exploration:
1. How do you factor a quadratic equation like \( x^2 + 2x - 8 = 0 \) without using the quadratic formula?
2. What happens if there are no real solutions for \( x \) when solving a system of equations?
3. How would you solve a system of equations graphically?
4. What is the geometric interpretation of the solutions to a system of two equations?
5. Can this system of equations be solved using matrix methods?

**Tip:** Always check your solutions by plugging them back into the original equations to verify they satisfy both equations.

Solve the system of equations: xy = 16 and 2x - y = -4

Learn how to solve the system of equations xy = 16 and 2x - y = -4 using substitution and the quadratic formula. This detailed step-by-step solution involves solving a quadratic equation and finding multiple solutions. Suitable for students in Grades 9-10.

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