Question

If $x+y=4$, then $2x+2y$ is: (A) 4 (B) 2 (C) 8 (D) 16 (E) Cannot be determined

Short answer

The value of $2x+2y$ is 8, so the answer is (C).

Step-by-step solution

Write the given equation

The given equation is: $$x+y=4$$

Analyze the expression to solve

We need to determine the value of $2x+2y$.

Factor the expression

Notice that $2x+2y$ can be factored by factoring out 2: $$2x+2y=2(x+y)$$

Substitute the known value

From the given equation $x+y=4$, substitute 4 into the factored expression: $$2(x+y)=2\times 4$$

Simplify the expression

Simplify the multiplication: $$2\times 4=8$$

Conclude the result

Thus, the value of $2x+2y$ is 8, corresponding to option (C).

Key concepts

algebra problems

Algebra problems are fundamental exercises to test your mathematical skills and enhance your problem-solving abilities. To tackle algebra problems efficiently, you need to comprehend the question requirements, simplify terms, and carry out operations methodically. In our exercise, the given equation $x+y=4$ is used to determine another expression. Break down the expression into manageable parts. For example, the problem asked us to find the value of $2x+2y$. By understanding that algebra is about manipulating expressions to reveal unknown values, you can begin to see how every part of the equation is interrelated. Always start by writing down what is given, and then transform and solve step-by-step.

factoring expressions

Factoring expressions is a crucial skill in algebra that involves rewriting expressions as products of simpler factors. It simplifies complex problems and makes solving equations easier. In this exercise, we transformed $2x+2y$ by factoring out the common term. Notice that both terms in $2x+2y$ share a common factor of 2. By factoring 2 out, the expression becomes $2(x+y)$. This allowed us to substitute the known value $x+y=4$, making it straightforward to calculate the result. Always look for common factors in terms within an expression to simplify and factor effectively. This helps to solve problems more efficiently and reduce potential errors.

solving equations

Solving equations involves finding the value of unknown variables that make the equation true. For linear equations, it often means isolating the variable or variables. For the exercise provided, we needed to find the value of the expression $2x+2y$ given the equation $x+y=4$. First, recognize the need to manipulate the equation for substitution. We factored $2x+2y$ to get $2(x+y)$. Then, we substituted $x+y=4$ into the factored form, yielding $2\times 4$. Finally, solving this gives us 8. The logical steps of substitution, simplification, and solving ensure you reach the correct answer. Practicing these steps helps you become proficient in solving more complex algebraic equations.

mathematics for GMAT

Mathematics for GMAT includes a wide range of topics like algebra, arithmetic, and geometry. Mastery of these areas is necessary for success. For the GMAT, it is vital to develop a strong foundation in core mathematical concepts and problem-solving techniques. In our algebra problem, we used fundamental principles to algebraically manipulate and simplify expressions. Such problems test your ability to apply mathematical operations and understanding quickly and accurately. Efficient problem-solving strategies such as factoring, substitution, and simplifying expressions are crucial. Additionally, practicing varied problems enhances speed and precision, skills necessary for the GMAT. Always review each solution methodically to ensure all steps logically follow one another, maintaining accuracy. Regular practice with similar problems helps in reinforcing these concepts.