Question

What is the value of \(x y ?\) (1) \(x+y=6\) (2) \(2 x+y=10\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

C. 1 and 2 together (need both)

Step-by-step solution

- Analyze statement 1

Statement 1 tells us that \(x + y = 6\). This equation alone is not enough information to determine the specific values of \(x\) and \(y\), and therefore we cannot determine the value of \(xy\) from this statement alone.

- Analyze statement 2

Statement 2 tells us that \(2x + y = 10\). This equation alone is also not enough to determine the specific values of \(x\) and \(y\), so we cannot find the value of \(xy\) from this statement alone either.

- Combine both statements

Now combine the two equations: \(x + y = 6\) and \(2x + y = 10\). Subtract the first equation from the second:\r\[(2x + y) - (x + y) = 10 - 6\]\rThis simplifies to:\r\[x = 4\]\rSubstitute \(x = 4\) back into the first equation:\r\[4 + y = 6\]\rSo, \(y = 2\). Now, we can find the product \(xy\):\r\[xy = 4 \times 2 = 8\]

Key concepts

GMAT equations

In this exercise, we are given two linear equations and asked to determine the product of two variables, specifically: What is the value of \(xy\)? Linear equations are the building blocks of algebra and crucial for GMAT problem solving.
To solve these equations, we need to recognize whether a single equation is enough or if we need to combine multiple equations.
In this case, statement 1 gives us \(x + y = 6\). While this highlights the sum of \(x\) and \(y\), it does not tell us their individual values. The equation is therefore not enough on its own.

When we examine statement 2, \(2x + y = 10\), we face a similar problem. This equation alone won't help us find specific values for \(x\) and \(y\).

Combining both equations, \(x + y = 6\) and \(2x + y = 10\), gives us the ability to solve for individual values:

• Subtract statement 1 from statement 2: \((2x + y) - (x + y) = 10 - 6\)
• This simplifies to \(x = 4\)
• Insert \(x = 4\) into the first equation: \(4 + y = 6\)

Solving this yields \(y = 2\). Now that we have both values, \(xy\) is simply \(4 \times 2 = 8\).

algebraic expressions

Understanding and manipulating algebraic expressions is another key skill. Algebraic expressions like \(x + y = 6\) or \(2x + y = 10\) are foundational to solving more complex problems.
With algebraic expressions, we can represent mathematical ideas and relationships concisely. This problem involves linear expressions with two variables.
Decomposing the second equation using the given expressions:

• Subtract the first equation from the second
• Simplify remaining terms to isolate a variable
• Solve or substitute back into another equation to find each variable's value

This exercise allows students to see how varying expressions relate to each other in solving a question.

test preparation

Effective test preparation is paramount for handling GMAT problems. When faced with equations, students should practice identifying what each statement contributes.
Step-by-step methods are crucial:

• Assess whether a single equation can solve for variables or if more info is needed
• Determine how equations can be combined
• Practice different types of algebraic manipulations

Lastly, consistent problem-solving practice ensures the skills needed for exams are sharp. Understanding fundamental concepts in equations not only aids in specific questions but builds a strong foundation for more complex problem-solving.