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Solving Linear Equations

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This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn:

  • Equivalent Equations
  • Solving Linear Equations
  • Isolate the Variables
    • Examples with Single Step
    • Examples with Multiple Steps
    • Examples using Combining Like Terms
    • Examples using Distributive Property

Equivalent Equations

To solve an equation means to find the values of the variables that make the equation true; that is, the values that satisfy the equation.

Equivalent equations are equations that have the same solution.

For example, 2x + 3 = 11 and 2x = 11 − 3 and are equivalent equations; both are true when x = 4 and are false otherwise. The general method for solving an equation is to find successively simpler equivalent equations so that the simplest equivalent equation makes the solutions obvious.

The following rules are important for producing equivalent equations.
• When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation.
• When both sides of an equation are multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation.

For example, the following equations are equivalent to x = 6.


Equation Equivalent to x = 6

Add 3 to both sides

x + 3 = 6 + 3
x + 3 = 9

Subtract 2 from both sides

x – 2 = 6 – 2
x – 2 = 4

Multiply by 4 on both sides

4x = 6 × 4
4x = 24

Divide 2 on both sides

\(\begin{array}{l}\frac{x}{2} = \frac{6}{2}\\x = 3\end{array}\)

Solving Linear Equations

We can use equivalent equations to solve an equation. The solution is obtained when the variable is by itself on one side of the equation. The objective, then, is to use equivalent equations to isolate the variable on one side of the equation.

Consider the equation x + 6 = 14.
For the equation to be considered solved, the x has to be on a side by itself. How can we get rid of the + 6 so that x can be by itself? We can do the opposite operation i.e. to subtract 6. In order to get an equivalent equation, we could subtract 6 from both sides.

x + 6 = 14
x + 6 − 6 = 14 − 6
x = 8

We can check the solution to be sure that it is correct.

x + 6 = 14
8 + 6 = 14
14 = 14

The following videos show how to isolate the variable in order to solve an equation.

Examples using Multiple Steps

Sometimes, we may require multiple steps to solve an equation.

The following video shows how to solve equations that require multiple steps.

The following two videos show how to solve equations by combining like terms and using the addition property of equality. 

Examples using Distributive Property

If there are parenthesis in the equation, we may first need to use the distributive property to remove the parenthesis before isolating the variable.

The following video shows some examples of solving equations that involve the distributive property and combining like terms.

The following video shows some examples of solving equations that involve the distributive property when there is a negative sign. The following video shows some examples that involve distributing, combining like terms, and using the addition property, with attention given to the negative signs, and particular case of distributing a negative 1. 

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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