Math Problem Statement

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

Solution

The system of equations given in the image is:

xy=16xy = 16 2xy=42x - y = -4

We can solve this system step by step.

Step 1: Solve for yy in the second equation

The second equation is 2xy=42x - y = -4. We can solve for yy in terms of xx: y=2x+4y = 2x + 4

Step 2: Substitute y=2x+4y = 2x + 4 into the first equation

Now substitute this expression for yy into the first equation xy=16xy = 16: x(2x+4)=16x(2x + 4) = 16

Step 3: Expand and solve the quadratic equation

Expanding the equation gives: 2x2+4x=162x^2 + 4x = 16 Now, bring all terms to one side: 2x2+4x16=02x^2 + 4x - 16 = 0 Divide through by 2 to simplify: x2+2x8=0x^2 + 2x - 8 = 0

Step 4: Solve the quadratic equation

This quadratic equation can be solved using the quadratic formula: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Here, a=1a = 1, b=2b = 2, and c=8c = -8. Plugging these values into the quadratic formula: x=2±224(1)(8)2(1)x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)} x=2±4+322x = \frac{-2 \pm \sqrt{4 + 32}}{2} x=2±362x = \frac{-2 \pm \sqrt{36}}{2} x=2±62x = \frac{-2 \pm 6}{2} Thus, the two solutions for xx are: x=2+62=2orx=262=4x = \frac{-2 + 6}{2} = 2 \quad \text{or} \quad x = \frac{-2 - 6}{2} = -4

Step 5: Find corresponding yy values

Now, substitute each xx value into y=2x+4y = 2x + 4 to find the corresponding yy:

  • When x=2x = 2: y=2(2)+4=4+4=8y = 2(2) + 4 = 4 + 4 = 8
  • When x=4x = -4: y=2(4)+4=8+4=4y = 2(-4) + 4 = -8 + 4 = -4

Final Solution

The solutions to the system are: (x,y)=(2,8)or(x,y)=(4,4)(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 x2+2x8=0x^2 + 2x - 8 = 0 without using the quadratic formula?
  2. What happens if there are no real solutions for xx 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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Quadratic Equations

Formulas

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Substitution method

Theorems

Quadratic Formula Theorem

Suitable Grade Level

Grades 9-10