Solution of Equation
Solution of Equation
The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true. For equations having one unknown, raised to a single power, two fundamental rules of algebra, including the additive property and the multiplicative property, are used to determine its solutions. Solutions for equations with multiple unknown variables are found by using the principles for a system of equations. Equations with terms raised to a power greater than one can be solved by factoring and, in some specific cases, by the quadratic equation.
The idea of a solution of equations has existed since the time of the ancient Egyptians and Babylonians. During these times, they used simple algebraic methods to determine solutions for practical problems related to their everyday life. The methods used by the ancients were preserved in a treatise written by Arabian mathematician Al-Kowarizmi AD 825). In this work, he includes methods for solving linear equations as well as second-degree equations. Solutions for some higher degree equations were worked out during the sixteenth century by Italian mathematician Gerolamo Cardano (1501–1576).
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 that can be substituted into the equation to make it a true statement. For the first example, the solution for x is 13. The second example has two values that 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 = 7has a solution of x = 3. According to the first rule, one 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 the 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 – 4orx = 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, one can obtain an equivalent equation, one with the same solution set, by multiplying both sides by any number, such as 2. Thusthe equation becomes2y– 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.
Most equations are given in a more complicated form that can be simplified. Consider the equation 4x – x – 5 = 2x + 7. The first step in solving this equation is to combine like terms on each side of the equation. On the right side, there are no like terms, but the 4x and –x on the left side are like terms. This equation, when simplified, becomes 3x – 5 = 2x + 7. The next step is to eliminate the unknown from one side of the equation. For this example, this is accomplished by adding –2x to both sides of the equation, which gives x – 5 = 7. Using the additive property, the solution is obtained by adding 5 to both sides of the equation, so x = 12.
The whole process for solving single variable algebraic equations can be summarized by the following steps. First, eliminate any parentheses by multiplying out factors. Second, add the like terms in each side. Third, eliminate the unknown from one side of the equation using the multiplicative or additive properties. Fourth, eliminate the constant term from the side with the unknown using the additive property. Finally, eliminate any coefficient on the unknown by using the multiplicative property.
Many algebraic equations contain more than one variable, so the complete solution set can not be found using the methods described thus far. Equations with two unknowns are called linear equations and can be represented by the general formula ax + by = c; where a, b, and c are constants and x and y are variables. The solution of this type of equation would be the ordered pair of x and y that makes the equation true. For example, the solution set for the equation x + y = 7 would contain all the pairs of values for x and y that satisfy the equation, such as (2,5), (3,4), (4,3), etc. In general, to determine the solution to a linear equation with two variables, the equation is rewritten and solved in terms of one variable. The solution for the equation x + y = 7, then becomes any pair of values that makes x = 7 – y true.
Often multiple linear equations exist which relate two variables in the same system. All of the equations related to the variables are known as a system of equations and their solution is an ordered pair that makes every equation true. These equations are solved by methods of graphing, substitution, and elimination.
Equations that involve unknowns raised to a power of one are known as first-degree equations. Second-degree equations also exist which involve at
Additive property —The property of an equation that states a number can be added to both sides of an equation without affecting its solution.
Factoring —A method of reducing a higher degree equation to the product of lower degree equations.
First degree equation —An algebraic expression that contains an unknown raised to the first power.
Multiplicative property —The property of an equation that state all the terms in an equation can be multiplied by the same number without affecting the final solution.
Second degree equation —An algebraic expression that contains an unknown raised to the second power.
least one variable that is squared or raised to a power of two. Equations can also be third-degree, fourth-degree, and so on. The most famous second-degree equation is the quadratic equation, which has the general form ax2 +bx +c = 0; where a, b, and c are constants and a is not equal 0. The solution for this type of equation can often be found by a method known as factoring.
Since the quadratic equation is the product of two first-degree equations, it can be factored into these equations. For example, the product of the two expressions (x + 2)(x – 3) provides one with the quadratic expression x2 – x – 6. The two expressions (x + 2) and (x – 3) are called factors of the quadratic expression x2 – x – 6. By setting each factor of a quadratic equation equal to zero, solutions can be obtained. In this quadratic equation, the solutions are x =–2 and x = 3.
Finding the factors of a quadratic equation is not always easy. To solve this problem, the quadratic formula was invented so that any quadratic equation can be solved. The quadratic equation is stated as follows for the general equation ax2 + bx + c = 0
To use the quadratic formula, numbers for a, b, and c are substituted into the equation, and the solutions for x are determined.
See also Systems of equations.
Bittinger, Marvin L, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 7th ed. Reading, MA: Addison-Wesley Publishing, 2006.
Larson, Ron. Precalculus. 7th ed. Boston, MA: Houghton Mifflin, 2007.
Lorenz, Falko. Algebra. New York: Springer, 2006.
Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.