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(x-8)(y-9)=0 Y-5/x+y-13=4

3 min read Jun 17, 2024
(x-8)(y-9)=0 Y-5/x+y-13=4

Solving a System of Equations: (x-8)(y-9) = 0 and y-5/x+y-13=4

This article will guide you through the process of solving a system of equations involving one equation with a product and one equation with a fraction.

Understanding the Equations

Equation 1: (x-8)(y-9) = 0

This equation tells us that either (x-8) = 0 or (y-9) = 0. This is because the product of two numbers is zero only if at least one of them is zero.

Equation 2: y-5/x+y-13=4

This equation involves a fraction with x in the denominator. To solve this, we need to get rid of the fraction.

Solving the System

  1. Solve Equation 1 for x and y:

    • If (x-8) = 0, then x = 8.
    • If (y-9) = 0, then y = 9.

  2. Substitute the values of x and y into Equation 2:

    • Case 1: x = 8
      • Substitute x = 8 into Equation 2: y - 5/8 + y - 13 = 4
      • Simplify and solve for y: 2y - 109/8 = 4 2y = 137/8 y = 137/16
    • Case 2: y = 9
      • Substitute y = 9 into Equation 2: 9 - 5/x + 9 - 13 = 4
      • Simplify and solve for x: 5 - 5/x = 4 -5/x = -1 x = 5

  3. Check the Solutions:

    • Solution 1: (x, y) = (8, 137/16)
      • Substitute these values back into the original equations to verify they satisfy both equations.
    • Solution 2: (x, y) = (5, 9)
      • Substitute these values back into the original equations to verify they satisfy both equations.

The Solution

The system of equations has two solutions:

  • (x, y) = (8, 137/16)
  • (x, y) = (5, 9)

Remember to always check your solutions by plugging them back into the original equations.

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