Solution of Equation

An equation is an algebraic expression which typically relates unknown variables to other variables or constants. For example, x + 2 = 15 is an equation, as is y²= 4. The solution, or root, of an equation is any value or set of values which can be substituted into the equation to make it a true statement. For our first example, the solution for x is 13. The second example has two values which will make the statement true, namely 2 and -2. These values make up the solution set of the equation.

Using the two fundamental rules of algebra, solutions to many simple equations can be obtained. The first rule states that the same quantity can be added to both sides of an equation without changing the solution to the equation. For example, the equation x + 4 = 7 has a solution of x = 3. According to the first rule, we can add any number to both sides of the equation and still get the same solution. By adding 4 to both sides, the equation becomes x + 8 = 11 but the solution remains x = 3. This rule is known as the additive property of equality. To use this property to find the solution to an equation, all that is required is choosing the right number to add. The solution to our previous example x + 4 = 7 can be found by adding -4 to both sides of the equation. If this is done, the equation simplifies to x + 4 - 4 = 7 - 4 or x = 3 and the equation is solved.

The second fundamental rule, known as the multiplicative property of equality, states that every term on both sides of an equation can be multiplied or divided by the same number without changing the solution to the equation. For instance, the solution for the equation y - 2 = 10 is y = 12. Using the multiplicative rule, we can obtain an equivalent equation, one with the same solution set, by multiplying both sides by any number, such as 2. Thus the equation becomes 2y - 4 = 20, but the solution remains y = 12. This property can also be used to solve algebraic equations. In the case of the equation 2x = 14, the solution is obtained by dividing both sides by 2. When this is done 2x/2 = 14/2 the equation simplifies to x = 7.

Often, both of these rules must be employed to solve a single equation, such as the equation 4x + 7 = 23. In this equation, -7 is added to both sides of the equation and it simplifies to 4x = 16. Both sides of this equation are then divided by 4 and it simplifies to the solution, x = 4.