Math Problem Statement

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

Solution

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

Step 1: Rearrange the equation of the line

First, express yy from the equation of the line in terms of xx: y=112xy = 11 - 2x

Step 2: Substitute into the equation of the curve

Now, substitute y=112xy = 11 - 2x into the curve equation xy=12xy = 12: x(112x)=12x(11 - 2x) = 12

Step 3: Simplify and solve for xx

Expanding the equation: 11x2x2=1211x - 2x^2 = 12 Rearrange the terms: 2x211x+12=02x^2 - 11x + 12 = 0

Step 4: Solve the quadratic equation

Now solve the quadratic equation 2x211x+12=02x^2 - 11x + 12 = 0 using the quadratic formula: x=(11)±(11)24(2)(12)2(2)x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(12)}}{2(2)} x=11±121964x = \frac{11 \pm \sqrt{121 - 96}}{4} x=11±254x = \frac{11 \pm \sqrt{25}}{4} x=11±54x = \frac{11 \pm 5}{4}

So, the two solutions for xx are: x1=11+54=4andx2=1154=1.5x_1 = \frac{11 + 5}{4} = 4 \quad \text{and} \quad x_2 = \frac{11 - 5}{4} = 1.5

Step 5: Find the corresponding yy values

For x1=4x_1 = 4, substitute into y=112xy = 11 - 2x: y1=112(4)=118=3y_1 = 11 - 2(4) = 11 - 8 = 3

For x2=1.5x_2 = 1.5, substitute into y=112xy = 11 - 2x: y2=112(1.5)=113=8y_2 = 11 - 2(1.5) = 11 - 3 = 8

Step 6: Conclusion

The points of intersection are: (4,3)and(1.5,8)(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.

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Equations
Quadratic Equations

Formulas

Equation of a line: y = mx + b
Quadratic equation formula: ax^2 + bx + c = 0

Theorems

Quadratic formula

Suitable Grade Level

Grades 9-11