Question

If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.

Hint:

The principle of equality of ordered pairs is fundamental and extends to vectors and matrices. Always equate corresponding elements to form separate, simpler equations to solve.

Answer 1

Step 1: Understanding the Concept:
Two ordered pairs (a, b) and (c, d) are considered equal if and only if their corresponding components are equal. This means that a must be equal to c, and b must be equal to d.
Step 2: Key Formula or Approach:
Given the equality of ordered pairs (2x - 3, 5) = (x, y - 1), we set up a system of two linear equations by equating the first components and the second components separately.
1. First component equation: \(2x - 3 = x\)
2. Second component equation: \(5 = y - 1\)
Step 3: Detailed Explanation:
We solve the two equations to find the values of x and y.
Solving for x: \[ 2x - 3 = x \] Subtract x from both sides: \[ 2x - x - 3 = 0 \] \[ x - 3 = 0 \] Add 3 to both sides: \[ x = 3 \] Solving for y: \[ 5 = y - 1 \] Add 1 to both sides: \[ 5 + 1 = y \] \[ y = 6 \] Step 4: Final Answer:
The values are x = 3 and y = 6.