Question

If \(2 x-6 y=9\), then \(3 y-x\) is: A. 4.5 B. -4.5 C. 9 D. 3 E. Cannot be determined

Short answer

-4.5

Step-by-step solution

- Rewrite the Equation

Start with the given equation: 2x - 6y = 9. Rewrite it to express one variable in terms of the other. Let's solve for x in terms of y. 2x = 6y + 9 x = 3y + 4.5

- Substitute x into the Expression

Now substitute the expression for x into the expression we need to find: 3y - x. Since x = 3y + 4.5, we have: 3y - (3y + 4.5)

- Simplify the Expression

Simplify the expression: 3y - 3y - 4.5 The terms 3y cancel each other out, leaving: -4.5

Key concepts

Algebra

Understanding algebra is crucial for solving problems on the GMAT. It involves using symbols, usually letters, to represent numbers in equations. These symbols can be manipulated using the rules of arithmetic. For example, in the exercise given, the equation is written as: \(2x - 6y = 9\). We need to isolate one variable to solve for the desired expression. Remember that algebra relies on two main principles: balancing equations and performing the same operation on both sides of the equation. This is the foundation for more complex problem-solving techniques.

Linear Equations

A linear equation represents a straight line when graphed. In the form \(ax + by = c\), it can be solved for one variable in terms of another. Take the equation given in the exercise: \(2x - 6y = 9\). The aim is to simplify this into the form \(x = 3y + 4.5\), a typical algebraic manipulation where we rearrange the equation to isolate x. Linear equations are fundamental in analyzing relationships between two variables and are frequently tested in GMAT problems. They form the basis for solving many real-world scenarios involving constant rates and ratios.

Substitution Method

The substitution method is a reliable technique to solve systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. In our exercise, we first solve for x: \(x = 3y + 4.5\). Next, we substitute this into the expression \(3y - x\) that we need to simplify. Substitution simplifies the problem by reducing the number of variables, making it easier to solve. This method is especially useful when equations are easily manipulated to isolate one variable, as shown in the steps provided in the solution.

Simplification

Simplification is the process of making an expression easier to handle. It involves combining like terms and reducing the expression to its simplest form. In the given solution, after substituting \(x = 3y + 4.5\) into \(3y - x\), we get \(3y - (3y + 4.5)\). By distributing and combining like terms, this simplifies to \(-4.5\). This step is crucial in solving equations as it helps in reaching the final, simplified answer. Simplification reduces the complexity of calculations and ensures that solutions are presented in their most concise form.