Introduction to Solving Equations

Let’s review some background material that you’ll need to study equation solving.

Definition of an Equation

An equation is a statement of equality between two expressions separated into a left and right side by an equal sign (=). Inequalities are algebraic relations showing that a quantity is greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤) another quantity.

What It Means to Solve an Equation

Finding a solution to an equation means finding the set of all x values that satisfy the equation when substituted into it. With this in mind, to check that a given value is indeed a solution, substitute it into the equation and make sure that the left-hand side (LHS) of the equation has the same value as the right-hand side (RHS).

Example: Solve for x in 3x – 5 = 1 and check your answer.

Solution:

      3x – 5 = 1

3x – 5 + 5 = 1 + 5

              3x = 6

          3x/3 = 6/3

                 x = 2

Now, let’s check that x = 2 is indeed a solution by verifying that LHS = RHS

LHS                            RHS

3(2) - 5                           1

= 6 - 5

= 1

Since LHS = RHS, we conclude that 2 is indeed a solution.

What it means to find a solution to an inequality

A solution to an inequality means finding the set of all x values that satisfy the required inequality when substituted in.

Example: Solve for the set of x that satisfies√x - 2  > 10.

  √ x - 2   > 10

(√ x - 2  )² > 10²

         x – 2 > 100

                x > 102

It’s a bit harder to check our answer here, but you’ll notice that if you substitute in a value of x that’s greater than 102 (e.g., 146), the inequality should hold (i.e., be satisfied), but if you substitute in a value of x smaller than 102 (e.g., 83), the inequality isn’t satisfied. This doesn’t prove that your answer is right, but it does give you a bit more confidence in it, and it might help you to find out if your answer is wrong.

Collecting like terms

The first step in solving an equation is usually to collect like terms. This may include adding and subtracting terms from each side to get all like terms on one side of an equation. Once the like terms have been collected it becomes possible to solve an equation by isolating the variable using multiplication, division, or other operations, and to finally solve for the unknown.

Example: Solve 3x + 5 = 7 – x for x.

Solution:

     3x + 5 = 7 – x

3x + 5 - 5 = 7 – x – 5

             3x = 2 – x

       3x + x = 2 – x + x

             4x = 2

         4x/4 = 2/4

                x = 1/2