find the coordoinates of the points of intersection of the line y + 2x + 11 and the curve xy = 12

To find the points of intersection of the line \( y + 2x = 11 \) and the curve \( xy = 12 \), we need to solve this system of equations.

### Step 1: Rearrange the equation of the line
First, express \( y \) from the equation of the line in terms of \( x \):
\[
y = 11 - 2x
\]

### Step 2: Substitute into the equation of the curve
Now, substitute \( y = 11 - 2x \) into the curve equation \( xy = 12 \):
\[
x(11 - 2x) = 12
\]

### Step 3: Simplify and solve for \( x \)
Expanding the equation:
\[
11x - 2x^2 = 12
\]
Rearrange the terms:
\[
2x^2 - 11x + 12 = 0
\]

### Step 4: Solve the quadratic equation
Now solve the quadratic equation \( 2x^2 - 11x + 12 = 0 \) using the quadratic formula:
\[
x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(12)}}{2(2)}
\]
\[
x = \frac{11 \pm \sqrt{121 - 96}}{4}
\]
\[
x = \frac{11 \pm \sqrt{25}}{4}
\]
\[
x = \frac{11 \pm 5}{4}
\]

So, the two solutions for \( x \) are:
\[
x_1 = \frac{11 + 5}{4} = 4 \quad \text{and} \quad x_2 = \frac{11 - 5}{4} = 1.5
\]

### Step 5: Find the corresponding \( y \) values
For \( x_1 = 4 \), substitute into \( y = 11 - 2x \):
\[
y_1 = 11 - 2(4) = 11 - 8 = 3
\]

For \( x_2 = 1.5 \), substitute into \( y = 11 - 2x \):
\[
y_2 = 11 - 2(1.5) = 11 - 3 = 8
\]

### Step 6: Conclusion
The points of intersection are:
\[
(4, 3) \quad \text{and} \quad (1.5, 8)
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you solve quadratic equations by factoring?
2. What are the general methods to solve systems of equations?
3. How do you check the solutions of a system of equations?
4. What is the importance of the quadratic formula?
5. Can every system of equations be solved using substitution?

**Tip:** Always check your solutions by substituting them back into the original equations.

find the coordinates of the points of intersection of the line y + 2x + 11 and the curve xy = 12

Learn how to find the points of intersection of the line y + 2x = 11 and the curve xy = 12. This problem involves solving a system of equations using substitution and applying the quadratic formula. Suitable for students in Grades 9-11.

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