# Linear Equations - GMAT Math Study Guide

## Definitions

• Linear Equation - An equation involving variables of the first degree only (i.e., no quadratic (x2) or cubic (x3) terms).
For example, 10x + 2 = 5x + 12 is a linear equation while x2 + 4 = x - 5 is not a linear equation but is rather a quadratic equation.
• Solution - A number that, when substituted for the unknown, will make the equation true.
For example, in the equation x + 3 = 4, the number 1 is the solution since 1 + 3 = 4

### Examples of Linear and Non Linear Equations

#### Linear Equations

The following is a linear equation, even though it has two different variables.

5x + 7y = 14

The following is a system of linear equations, even though it has two different equations with two different variables in each equation.

10x + 15y = 72
22x + 3y = 102

#### Non Linear Equations

The following are not linear equations since they have an exponent with a power that is not 1.

x2 - 2x + 1 = 0
x-5 + y-5 + 15x2 + 2 = 0

## Simple Linear Equations

You must perform the same operations on both sides of the equation.

A simple linear equation is one with a single variable. In this case, solving for the unknown variable is accomplished by isolating that unknown. When performing mathematical operations, remember that the same operation that is done to one side of the equation must be done to the other side of the equation. For example, if you add 5 to the left side of the equation, you must add 5 to the right side of the equation.

5x + 20 = 2x - 10
5x - 2x + 20 = 2x - 2x - 10
3x + 20 = - 10
3x + 20 - 20 = - 10 - 20
3x = - 30
x = -10

10x + 15 - 5x + 30 = 0
5x + 45 = 0
5x + 45 - 45 = 0 - 45
5x = -45
x = -9

## Types of GMAT Problems

1. Solving One Variable Linear Equations

Solving a linear equation involves isolating the variable on one side of the equation. To do so, use mathematical operations on both sides of the equation. Adding or subtracting the same numbers from both sides does not change the equation. Similarly, the equation will remain unchanged after multiplying or dividing both sides of the equation by the same number.

Find the value of x that satisfies the following equation: A) x=1 B) x=5 C) x=1/4 D) x=1/5 E) x=3
1. Multiply both sides by 3.
27x - 9 = 2x-4.
2. Add 9 to both sides.
27x = 2x + 5.
3. Subtract 2x from both sides.
25x = 5.
4. Divide both sides by 25.
x= (5/25) = (1/5)
2. Finding Solutions to Multi-Variable Linear Equations

If there are two variables, pick one variable and choose a number for it to equal. Substitute the chosen number into the equation and solve for the second variable. For equations with more than two variables, choose numbers for every variable but one. Substitute all of the numbers in and solve for the final variable.

Which of the following is NOT a solution to the equation 2x + 3y = 2y - 5x + 10?
 A) x=1, y=3 B) x=7, y=-38 C) x=0, y=10 D) x=5, y=-25 E) x=2, y=-4