` x = 2y + 5 `
` y = ( 2 x - 3 ) ( x + 9 ) `
How many ordered pairs ` ( x, y ) ` satisfy the system of equations shown above ?

Options

A. 0
B. 1
C. 2
D. Infinitely many

Explanation

To solve the system of equations given by:

1. \( x = 2y + 5 \)
2. \( y = (2x - 3)(x + 9) \)

we need to find how many ordered pairs \( (x, y) \) satisfy both equations.

### Step 1: Substitute the expression for \( y \) into the first equation

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = \frac{x - 5}{2} \]

Now, we substitute this expression for \( y \) into the second equation:
\[ \frac{x - 5}{2} = (2x - 3)(x + 9) \]

### Step 2: Clear the fraction

To eliminate the fraction, multiply both sides by 2:
\[ x - 5 = 2(2x - 3)(x + 9) \]

### Step 3: Expand the right side

Now, we need to expand the right side:
\[ x - 5 = 2[(2x)(x) + (2x)(9) - (3)(x) - (3)(9)] \]
\[ = 2[2x^2 + 18x - 3x - 27] \]
\[ = 2[2x^2 + 15x - 27] \]
\[ = 4x^2 + 30x - 54 \]

### Step 4: Rearrange the equation

Now we have:
\[ x - 5 = 4x^2 + 30x - 54 \]
Rearranging gives:
\[ 0 = 4x^2 + 30x - x - 54 + 5 \]
\[ 0 = 4x^2 + 29x - 49 \]

### Step 5: Use the quadratic formula to find \( x \)

The quadratic equation is \( 4x^2 + 29x - 49 = 0 \). We can find the discriminant \( D \) to determine the number of real roots:
\[ D = b^2 - 4ac = 29^2 - 4 \cdot 4 \cdot (-49) \]
\[ = 841 + 784 = 1625 \]

### Step 6: Analyze the discriminant

Since the discriminant \( D = 1625 \) is greater than 0, this means there are two distinct real roots for \( x \).

### Step 7: Find corresponding \( y \) values

For each value of \( x \), we can find a corresponding \( y \) using the equation \( y = \frac{x - 5}{2} \). Therefore, for the two distinct values of \( x \), we will have two corresponding values of \( y \).

### Conclusion

Thus, there are a total of **2 ordered pairs** \( (x, y) \) that satisfy the system of equations.

### Final Answer:
The number of ordered pairs \( (x, y) \) is **2**.

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