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:
$$ 2y + 4 + y = 10 $$

🔗 Related Articles You Might Like:

📰 Britney Spears Nude: What Really Happened Underneath the Shock? 📰 Britney Spears Nude Blast Exposes Secrets Only the Internet Could Unfold 📰 You Won’t Believe What These Breathe Into You Lyrics Reveal About Your Soul 📰 Rooftop Koreans Living Secret Lives Above The City Lights 📰 Rook Piercing That Guarantees Bold Spines You Wont Breathe Again 📰 Roose Harts Nudity Shocks The World In Disguise 📰 Roosevelt Avenue Link A Hidden Assassination Site Just Below Jackson Heights 📰 Roosevelt Avenue Subway Station Burned By Timethis Hidden Truth Shocked Local Experts 📰 Roosevelt Avenue Subway Station Where Whispers From The Past Still Echo Beneath Your Feet 📰 Roosevelt Avenues Hidden Dangers Beneath Jackson Heights Subway Station 📰 Roosevelt Avenues Secret Hub The Subway Station Where History Breaks Underground 📰 Rooss Hidden Price Drop Hides A 49 Cent Secret You Cannot Ignore 📰 Rooster Fish Drops Secrets From The Deep Thatll Blow Your Mind 📰 Rooster Fish Natures Surprising Protector Of Ancient Marine Secrets 📰 Roosterfish Revealedno One Saw This Monster Under The Sea Before It Struck 📰 Roosterfish Secrets The Deadly Grip No Deep Sea Diver Ever Documented 📰 Root Canal Sucks So Bad Youll Wish You Never Had It Done 📰 Roowe Furniture Hacks Every Homeowner Searches Foryoure Missing Them All

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.