Problem 50 - Algebra 1

Question

Solve the equation. $$ 2 y+5=3 y $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 5\).

Step 1: Simplifying the Equation

The first step is to simplify the given equation. In this case, it means moving the '2y' term from the left side of the equation to the right side to isolate 'y'. This is done by subtracting '2y' from both sides of the equation. \[ 2y + 5 - 2y = 3y - 2y \]

Step 2: Further Simplification

Subtracting '2y' from both sides simplifies the equation further to \[ 5 = y \]

Key Concepts

Equation Solving

Solving linear equations is a fundamental skill in algebra. It involves finding the value of an unknown variable that makes the equation true. For the equation \( 2y + 5 = 3y \), the goal is to find which value of \( y \) satisfies the equation.
To solve equations like this, you often need to perform operations that balance both sides, such as addition, subtraction, multiplication, or division. This ensures that the equation remains equivalent, helping you find the correct solution without altering the equation's meaning.

• Start by identifying what operations will isolate the variable.
• Make sure each step you take is valid by performing the same operation on both sides.

Solving equations requires practice, but understanding the logic behind each step is crucial for mastering this essential skill.

Isolating Variables

Isolating the variable is the key step in solving an equation. For the equation \( 2y + 5 = 3y \), we want to isolate \( y \) on one side of the equation. This means you need to rearrange the equation until the variable \( y \) stands alone.
When isolating variables, think of moving all terms containing the variable to one side and constant terms to the other. For example:

• Subtract \( 2y \) from both sides: \( 2y + 5 - 2y = 3y - 2y \).
• The result simplifies to \( 5 = y \).

This isolated form \( y = 5 \) clearly shows the value of \( y \) that satisfies the original equation. Understanding how to isolate variables effectively is foundational for solving equations.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic operations to transform and simplify equations. In the equation \( 2y + 5 = 3y \), algebraic manipulation helps rearrange the terms so we can solve for \( y \).
Here's how the steps look:

• Initially, get all terms involving \( y \) on one side: By subtracting \( 2y \) from both sides, you simplify the equation from \( 2y + 5 \) to \( 5 = 3y - 2y \).
• Simplify the right side further: \( 3y - 2y \) becomes \( y \).

These kinds of manipulations allow you to break down complex equations into simpler forms, making them easier to solve. Mastering algebraic manipulation develops your ability to think logically and solve problems efficiently.