Question

If \(x+y=2 x+2 y\), then \(4 x+4 y\) is: (A) 1 (B) 0 (C) 2 (D) 4 (E) Cannot be determined

Short answer

The value of \( 4x + 4y \) is 0, which corresponds to option (B).

Step-by-step solution

Identify and simplify the given equation

We have the equation: \[ x + y = 2x + 2y \] First, simplify by moving all terms to one side of the equation. Subtract \( x + y \) from both sides to get: \[ x + y - x - y = 2x + 2y - x - y \]}

Combine like terms

On combining like terms on both sides, we get:\[ 0 = x + y \] This implies that:\[ x + y = 0 \]

Find the value of the expression

We need to determine the value of \( 4x + 4y \). Using the equation \( x + y = 0 \), multiply both sides by 4 to get:\[ 4(x + y) = 4 \times 0 \]This simplifies to:\[ 4x + 4y = 0 \]

Determine the correct option

From the simplified form, \( 4x + 4y = 0 \). Therefore, the correct answer is (B) 0.

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols often represent numbers and are used to describe relationships and solve equations. In this problem, we use algebraic expressions and equations to find the desired value. Understanding how to manipulate these symbols is crucial for simplifying and solving algebraic problems efficiently.

Equation Simplification Made Easy

Equation simplification is about making an equation easier to solve. It involves combining like terms, removing unnecessary parts, and using basic arithmetic operations. Here, we start with the given equation:
\[ x + y = 2x + 2y \] We can move all terms to one side to get
\[ x + y - x - y = 2x + 2y - x - y \] This leads to a simplified form where we combine like terms, resulting in
\[ 0 = x + y \] Simplifying equations is a step-by-step process that makes it easier to solve and understand the problem.

Problem-Solving Steps

Effective problem-solving in math requires following a logical sequence of steps. Let's review the steps taken:

• Identify and simplify the given equation.


• Combine like terms on both sides of the equation.


• Use the simplified equation to find the required expression value.


• Determine the correct option based on your calculations.

Using these steps keeps you organized and ensures you don't miss any important details. It also helps you systematically break down complex problems into manageable parts.

Expression Evaluation

Evaluating expressions involves substituting known values into an expression and performing the operations to find the result. In this problem, after simplifying the equation to \[ x + y = 0 \] we need to find the value of \[ 4x + 4y \] We know that \[ x + y = 0 \] so we multiply both sides by 4:
\[ 4(x + y) = 4 \times 0 \] This simplifies to:
\[ 4x + 4y = 0 \] This step involves straightforward arithmetic, but it's crucial to understand the principles of substitution and operations when evaluating algebraic expressions.