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

SOLVED: Find the solution to the system of equations: x + 3y = 7 and 2x ...

Image Gallery

Solved: x+y+z=40 x=z+5 x=2y Solution Solve for the first variable in ...

Solved Solve the second-order equation xy''+2y'=12x2 by | Chegg.com

Solved - The surface given via the equation x2+y2=1−9z2 is | Chegg.com

Solved: X+y=8. X-y=2 elimination problem [Math]

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:

📰 You Won’t Believe What Happens When You Freeze Potatoes! 📰 Frozen Potatoes? Yes, They Work—Here’s How! 📰 Can You Actually Freeze Potatoes? Shocking Results You’ll Want to Try! 📰 Titan Mare Explorer 1488789 📰 Police Tips 6449872 📰 Cooking Simulator Vr 📰 Admirals Club Access 📰 Bloom Chicken 8250457 📰 Nvidia Stock Forum 📰 Cryptowatch 📰 Disney Hub Schedule Login 561836 📰 Unblocked Car Chase Games That Will Have You Swearing Driving Faster Than Ever 4615956 📰 Shocked Investors Heres Why Knsl Stock Is The Next Big Thingact Fast 9548414 📰 You Wont Dare To Unlock This Ziply Hacks Game Changer Anyone Is Using Now 2858927 📰 Master This 1 Tap Shortcut To Minimize Windows Instantly In 2024 1489627 📰 How To Boot Into Safe Mode Fastno Technical Skills Required 2127281 📰 597809 1204925 📰 Complete Your Christmas Decor In Seconds The Best Holiday Background Aesthetic Ideas 6939707

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.