Question

If \(2 x+3 y=20\) and \(2 x+3 z=20\), then \(x\) is: A. 4 B. 5 C. 2 D. 3 E. Cannot be determined

Short answer

E. Cannot be determined

Step-by-step solution

- Understand the given equations

The problem provides two equations: 1) 2x + 3y = 20 2) 2x + 3z = 20

- Compare the two equations

Notice that the left-hand side of both equations is 2x + 3(unknown variable), but the right-hand side is the same (20). Hence these two expressions are equal.

- Analyze what must be true for x

For the left-hand sides to be equal, the variables y and z do not affect the solution. This implies that any value of y or z would result in the same equation solving for x. Thus, just focus on the x-term.

- Isolate x

Since variations of y and z do not affect finding x: Solve for x in terms of one equation.For example,2x + 3y = 20.

- Solve for x

Isolate x: 2x = 20 - 3y x = (20 - 3y) / 2 Since there is no specific value for y, a numeric value for x cannot be directly determined.

Key concepts

solving equations

Solving equations is a fundamental skill in algebra and is essential for tackling various GMAT problems. The goal is to find the unknown variable that makes the equation true. When given an equation like \(2x + 3y = 20\), you need to follow a series of steps to isolate and solve for the variable. Start by understanding the structure of the equation and identify the variables and constants. You can manipulate the equation by performing the same operations on both sides until you've isolated the variable you're solving for. This process can involve adding, subtracting, multiplying, or dividing both sides of the equation.

variable isolation

Variable isolation is a critical step in solving algebraic equations. The aim here is to move all terms containing the variable of interest to one side of the equation and all other terms to the opposite side. For instance, given the equation \(2x + 3y = 20\), if you need to isolate \(x\), you start by removing terms involving \(y\). This implies subtracting \(3y\) from both sides, which gives: \(2x = 20 - 3y\). Next, you divide both sides by 2 to get: \(x = \frac{20 - 3y}{2}\). This approach helps simplify the equation so that you can explicitly define the variable.

equation comparison

Equation comparison is a useful technique when dealing with multiple equations. In this specific problem, we have two equations: \(2x + 3y = 20\) and \(2x + 3z = 20\). By comparing these two equations, you realize that the coefficient of \(x\) and the constants on the right-hand side are identical, which implies that the value of \(x\) is the same in both equations regardless of \(y\) and \(z\). This equality is key in simplifying your work and focusing solely on the variable of interest. Comparing each part of the equation allows you to deduce relationships and simplify your problem-solving process.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. Understanding how to manipulate these expressions is crucial in solving equations. Consider expressions like \(2x + 3y\) from the given problem. These expressions can be simplified or restructured to isolate the variables. For example, rearranging \(2x + 3y = 20\) allows you to solve for \(x\) once \(y\) is moved to the right side, resulting in: \(x = \frac{20 - 3y}{2}\). Mastery of algebraic expressions involves recognizing and applying the appropriate arithmetic operations to simplify and solve equations effectively.