To solve the equation \((y - 2, 2x + 1) = (x - 1, y + 2)\), we need to equate the corresponding components of the ordered pairs.

Given:
\[
(y - 2, 2x + 1) = (x - 1, y + 2)
\]

This implies two separate equations:
1. \(y - 2 = x - 1\)
2. \(2x + 1 = y + 2\)

Let's solve these equations step by step.

### Step 1: Solve the first equation
From the first equation:
\[
y - 2 = x - 1
\]
Add 2 to both sides:
\[
y = x + 1 \quad \text{(Equation 1)}
\]

### Step 2: Substitute \(y = x + 1\) into the second equation
Substitute the expression for \(y\) from Equation 1 into the second equation:
\[
2x + 1 = (x + 1) + 2
\]
Simplify:
\[
2x + 1 = x + 3
\]
Subtract \(x\) from both sides:
\[
x + 1 = 3
\]
Subtract 1 from both sides:
\[
x = 2
\]

### Step 3: Find \(y\)
Using \(x = 2\) in Equation 1:
\[
y = 2 + 1 = 3
\]

### Final Answer
The solution is:
\[
x = 2, \quad y = 3
\]

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How do you solve systems of equations by substitution?
2. What happens if the system of equations has no solution?
3. Can you solve this system graphically to verify the solution?
4. What is the elimination method, and how does it compare to substitution?
5. How can systems of equations be solved using matrices?

**Tip:** Always check your final solution by substituting the values back into the original equations to ensure they satisfy both equations.

Learn how to solve a system of equations by substitution with a detailed step-by-step solution. This example problem involves equating two ordered pairs and using the substitution method to find the values of x and y. Suitable for high school students studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation