Solving the Linear Equation: 2(y + 2) + 3y = 6

Understanding how to solve linear equations is a fundamental skill in algebra, essential for students, learners, and anyone wanting to strengthen their math foundation. One common type of equation encountered is in the form of a linear expression involving variables and constants—such as 2(y + 2) + 3y = 6.

This article breaks down the step-by-step process to solve this equation, explains the math concepts involved, and offers context on why mastering such equations matters.

Understanding the Context

What Is the Equation?

We begin with:
2(y + 2) + 3y = 6

This equation combines parentheses, variable terms, and constants. The goal is to find the value of y that makes the equation true.

Solved: 2) a) (2y+6)/y^2-9 / (3y^2+9y)/2y^2-6y [algebra]

Image Gallery

Solved: 2) a) (2y+6)/y^2-9 / (3y^2+9y)/2y^2-6y [algebra]
Solved 3y2-y-6y2+2y-8÷y2-9y2-3y+2= | Chegg.com
Solved y+2y-6+y2+5y+6y2-3y-18-y2-4y-12y2-12y+36 | Chegg.com
Solved 2/3 y" + 2y - 3y' = 3/4 y" - (6 + x) y - x x^3 y" + y | Chegg.com
Solved Do the following.(a) Factor 2y3-6y2-36y.(b) Solve | Chegg.com

Key Insights

Step-by-Step Solution

Step 1: Remove parentheses

Distribute the 2 across (y + 2):
2(y + 2) = 2y + 4

Now the equation becomes:
2y + 4 + 3y = 6

Step 2: Combine like terms

Combine the y terms:
2y + 3y = 5y

Continue Reading

Question: A historian studying the evolution of mathematical notation encounters an expression in a 17th-century manuscript:
Find the relationship between $ a $ and $ b $.
Thus, the only solution is when $ b = 0 $, and $ a $ is any nonzero real number. Therefore, the relationship is:

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Final Thoughts

Now the equation is:
5y + 4 = 6

Step 3: Isolate the variable term

Subtract 4 from both sides:
5y + 4 – 4 = 6 – 4
5y = 2

Step 4: Solve for y

Divide both sides by 5:
y = 2 ÷ 5
y = 0.4

Verification

Plug y = 0.4 back into the original equation:
2(0.4 + 2) + 3(0.4) = 2(2.4) + 1.2 = 4.8 + 1.2 = 6 ✅
The equation holds true.

Why Solving Linear Equations Matters

Linear equations like 2(y + 2) + 3y = 6 appear in real-world problems, including physics, economics, and engineering. Being able to:

  • Simplify expressions
  • Combine like terms
  • Isolate variables
  • Verify solutions

builds a strong foundation for more advanced math topics such as systems of equations, inequalities, and functions.