Question

If \(x y=1+2 x\) and \(2 x=1\), then \(y\) is: A. 4 B. 2 C. 1 D. \(8 / 3\) E. None of the above

Short answer

y is 4.

Step-by-step solution

Express x

Start by solving the second equation for x. The equation given is: \[2x = 1\]Dividing both sides by 2, we get:\[x = \frac{1}{2}\]

Substitute x in the first equation

Next, substitute the value of x into the first equation. The first equation is:\[xy = 1 + 2x\]Plugging in \(x = \frac{1}{2}\), we get:\[\left(\frac{1}{2}\right)y = 1 + 2\left(\frac{1}{2}\right)\]

Simplify the equation

Simplify the right-hand side of the equation:\[\left(\frac{1}{2}\right)y = 1 + 1\]\[\left(\frac{1}{2}\right)y = 2\]

Solve for y

Finally, solve for y by multiplying both sides by 2 to get rid of the fraction:\[y = 2 \times 2\]\[y = 4\]

Key concepts

algebraic equations

In algebra, an equation is a statement that asserts the equality of two expressions. There are different types of algebraic equations, such as linear equations and polynomial equations. The given GMAT problem, for example, involves solving linear equations to find the value of the variable y.

The first equation in the problem is: \[ xy = 1 + 2x \] which is a linear equation in terms of x and y.

In the second equation \[ 2x = 1 \], we can solve for x to simplify the first equation and find y. To solve algebraic equations efficiently, it's crucial to understand operations like addition, subtraction, multiplication, and division. These are used to isolate variables and solve for unknowns.

math test preparation

Preparing for the GMAT and other standardized tests requires a strategic approach. Here are some tips for mastering such problems:

• Practice Consistently: Work on a variety of problems regularly to build familiarity.
• Understand Concepts: Grasp the underlying concepts, such as properties of equations and operations with fractions.
• Time Management: Practice solving problems quickly to improve your time management skills during the test.
• Review Mistakes: Analyze errors to understand where you went wrong and to avoid repeating them.

For algebra problems, it’s important to be comfortable with basic operations, solving equations, and substitution. This ensures you can approach and solve problems efficiently on test day.

solving for variables

Solving for variables involves isolating the variable on one side of the equation. Let's break down how we did it in the given GMAT problem:

1. **Identify the equations**: \[ xy = 1 + 2x \] and \[ 2x = 1 \]

2. **Solve for x**: From the second equation, we divided both sides by 2:
\[ x = \frac{1}{2} \]

3. **Substitute x into the first equation**: Replace x in \[ xy = 1 + 2x \] with \[ \frac{1}{2} \]:
\[ \frac{1}{2}y = 1 + 2 \times \frac{1}{2} \] simplifies to:
\[ \frac{1}{2}y = 2 \]

4. **Solve for y**: Multiply both sides by 2 to isolate y:
\[ y = 4 \]

By isolating the variables step by step, we determine the final value of y.

step-by-step solutions

Step-by-step solutions are essential in solving complex math problems. They help in understanding the process and pinpointing where you might go wrong.

Here's the breakdown from our example:

1. **Express x**: Start with the easier equation, \[ 2x = 1 \]. Solve by dividing both sides by 2 to find \[ x = \frac{1}{2} \].

2. **Substitute x in the first equation**: Use the value of x in \[ xy = 1 + 2x \]. Replace x with \[ \frac{1}{2} \]:
\[ \frac{1}{2}y = 1 + 1 \]

3. **Simplify**: Simplify the equation to \[ \frac{1}{2}y = 2 \].

4. **Solve for y**: Multiply both sides by 2 to get \[ y = 4 \].

Such detailed steps ensure clarity and make it easier to follow the solution path logically, laying out each transformation until reaching the final answer.