Solution: Solve the system. From the second equation, $ x = y + 2 $. Substitute into the first: $ 2(y + 2) + y = 10 \Rightarrow 2y + 4 + y = 10 \Rightarrow 3y = 6 \Rightarrow y = 2 $. Then $ x = 2 + 2 = 4 $. The intersection is $ \boxed(4, 2) $.

Solve the System of Equations Step-by-Step: A Clear Guide to Finding the Solution

Solving a system of equations is a fundamental skill in algebra, essential for math students, engineers, and anyone working with real-world modeling. Understanding how to find the point where two equations intersect—also known as the solution to the system—is key to mastering linear relationships. In this article, we break down a classic linear system with clear steps to solve for $ x $ and $ y $.

Understanding the Problem

Understanding the Context

Suppose you’re given two equations:

$ x = y + 2 $
$ 2x + y = 10 $

These represent two lines on a coordinate plane. The goal is to find the single point $ (x, y) $ where both equations are true simultaneously—this is the solution to the system.

Step-by-Step Solution

Step 1: Substitute Using the Second Equation

$Solution: Solve the system. From the second equation, $ x = y + 2 $. Substitute into the first: $ 2(y + 2) + y = 10 \Rightarrow 2y + 4 + y = 10 \Rightarrow 3y = 6 \Rightarrow y = 2 $. Then $ x = 2 + 2 = 4 $. The intersection is $ \boxed{(4, 2)} $.$

Key Insights

We start with the second equation:
$$ 2x + y = 10 $$

We’re given a direct relationship between $ x $ and $ y $ in the first equation:
$$ x = y + 2 $$

Instead of solving for one variable at a time, substitution simplifies the process. Replace $ x $ in the second equation with $ y + 2 $:
$$ 2(y + 2) + y = 10 $$

Step 2: Expand and Simplify

Now expand the expression:
$$ 2y + 4 + y = 10 $$

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Final Thoughts

Combine like terms:
$$ 3y + 4 = 10 $$

Subtract 4 from both sides:
$$ 3y = 6 $$

Divide by 3:
$$ y = 2 $$

Step 3: Find $ x $ Using the First Equation

Now that we know $ y = 2 $, substitute back into the original equation $ x = y + 2 $:
$$ x = 2 + 2 = 4 $$

Step 4: Interpret the Solution

The solution to the system is the ordered pair where both equations meet:
$$ oxed{(4, 2)} $$

This means when $ x = 4 $ and $ y = 2 $, both equations are satisfied.

Why This Method Works

Substitution leverages the relationship between variables to reduce a system from two equations to one. This makes solving much simpler—especially helpful with harder systems involving more variables or nonlinear equations. It’s a foundational technique used across science and engineering fields.