Question

If \(x+y=2, x-y=98\), then \(x\) is: A. 50 B. 100 C. 98 D. 2 E. 102

Short answer

x is 50.

Step-by-step solution

Set up the equations

We are given two equations: 1. \(x + y = 2\)2. \(x - y = 98\)We need to solve these simultaneously to find the value of \(x\).

Add the equations

To eliminate \(y\), add the two equations together:\[ \begin{array}{rcl} x + y & = & 2 \ x - y & = & 98 \ \hline 2x & = & 100 \end{array}\]

Solve for \(x\)

Now solve for \(x\) by dividing both sides of the equation by 2:\[ 2x = 100 \ x = 50 \]

Key concepts

Linear Equations

A linear equation is an equation that makes a straight line when graphed. It has one or more variables with no exponents or powers attached to them. The general form of a linear equation is:
ax + b = c
where _a_, _b_, and _c_ are constants. In our exercise, we have two linear equations:

• \(x + y = 2\)
• \(x - y = 98\)

Each equation forms a line if plotted on a graph.
Linear equations are important in algebra because they set the foundation for more complex topics like quadratic equations and polynomials.

Simultaneous Equations

Simultaneous equations are a set of equations that have common variables. The goal is to find the values of these variables that satisfy all the equations in the set. In our exercise, we are given two simultaneous equations:

• \(x + y = 2\)
• \(x - y = 98\)

To solve simultaneous equations, we often use methods like substitution, elimination, or graphing. In the given solution, we use the elimination method. By adding the two equations:
\[x + y + x - y = 2 + 98\]
we eliminate _y_, leaving us with:
\[2x = 100\]
This simplifies to find the value of _x_. This method is efficient and straightforward for solving linear equations with two variables.

Basic Algebra

Basic algebra involves the manipulation of algebraic expressions and solving simple equations. It is the foundation of higher-level math topics. In our exercise, we use basic algebraic steps to find the value of _x_.
Here’s a breakdown of the steps:

• Set up the equations from the problem statement: \(x + y = 2\) and \(x - y = 98\).
• Add the equations to eliminate _y_: \(2x = 100\).
• Solve for _x_ by dividing both sides by 2: \(x = 50\).

Basic algebra techniques like combining like terms and isolating variables are used here. This problem helps strengthen your understanding of these fundamental concepts, setting the stage for tackling more complex algebra problems in the future.
Practicing such problems improves problem-solving skills and prepares you for standardized tests like the GMAT.