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:

📰 The Secret Tea Set That Makes Every Breakfast Feel Like a Luxury Retreat 📰 Discover the Hidden Tea Set No One Shares – It’s Extraordinary in Every Way 📰 How This Tea Set Cut Your Morning Routine in Half – Without Saying Yes 📰 The Magic Wave Of Good Vibeshappy Wednesday Like Never Before 📰 The Magical Dress That Made Tradition Look Like Fashion 📰 The Maiden Name Hinges On This Secret French Phrase Youve Never Noticed 📰 The Male Lion Fears Herthis Lioness Rewrote The Rules Of Dominance 📰 The Man Who Carries Gallons Holds Power Beyond Your Imagination 📰 The Man Who Screams While Sipping Tequilahell Knows What Hes Hiding 📰 The Massive Forearm Design Men Refuse To Hide 📰 The Massive Mistake Everyone Makes When Switching From Feet To Kilometers 📰 The Massive Weekly Ad That Shocks Every Viewer You Wont Believe What It Contains 📰 The Massive Weekly Ad Thats Covering Everything You Needsee For Yourself 📰 The Mic In Your Hand Ruins Everythingheres How Youll Regret Not Speaking Up 📰 The Milk In My Gallon Unlocked A Secret No One Wants You To Know 📰 The Mind Blowing Reality Of Gwasis Hidden Past You Need To See This 📰 The Mind Blowing Reason Floatmingo Changed The Internet Forever 📰 The Miracle Behind Giannis Immortalityyou Wont Believe His Hidden Power

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.