First, eliminate any parentheses by multiplying out factors. The whole process for solving single variable algebraic equations can be summarized by the following steps. Using the additive property, the solution is obtained by adding 5 to both sides of the equation, so x = 12. For this example, this is accomplished by adding –2x to both sides of the equation, which gives x – 5 = 7. The next step is to eliminate the unknown from one side of the equation. This equation, when simplified, becomes 3x – 5 = 2x + 7. On the right side, there are no like terms, but the 4x and –x on the left side are like terms. The first step in solving this equation is to combine like terms on each side of the equation. Consider the equation 4x – x – 5 = 2x + 7. Most equations are given in a more complicated form that can be simplified. Both sides of this equation are then divided by 4 and it simplifies to the solution, x = 4. In this equation, –7 is added to both sides of the equation and it simplifies to 4x = 16. Often, both of these rules must be employed to solve a single equation, such as the equation 4x + 7 = 23. When this is done 2x/2 = 14/2 the equation simplifies to x = 7. In the case of the equation 2x = 14, the solution is obtained by dividing both sides by 2. This property can also be used to solve algebraic equations. Thusthe equation becomes2y – 4= 20, but the solution remains 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. For instance, the solution for the equation y – 2 = 10 is y = 12. 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. If this is done, the equation simplifies to x + 4 – 4 = 7 – 4orx = 3 and the equation is solved. The solution to the previous example x + 4 = 7 can be found by adding –4 to both sides of the equation. To use this property to find the solution to an equation, all that is required is choosing the right number to add. This rule is known as the additive property of equality. By adding 4 to both sides, the equation becomes x + 8 = 11 but the solution remains x = 3. According to the first rule, one can add any number to both sides of the equation and still get the same solution. For example, the equation x + 4 = 7has a solution of x = 3. The first rule states that the same quantity can be added to both sides of an equation without changing the solution to the equation. Using the two fundamental rules of algebra, solutions to many simple equations can be obtained. These values make up the solution set of the equation. The second example has two values that will make the statement true, namely 2 and –2. For the first example, the solution for x is 13. 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 example, x + 2 = 15 is an equation, as is y 2 = 4. Methods for solving simple equationsĪn equation is an algebraic expression which typically relates unknown variables to other variables or constants. Solutions for some higher degree equations were worked out during the sixteenth century by Italian mathematician Gerolamo Cardano (1501 –1576). In this work, he includes methods for solving linear equations as well as second-degree equations. The methods used by the ancients were preserved in a treatise written by Arabian mathematician Al-Kowarizmi AD 825). During these times, they used simple algebraic methods to determine solutions for practical problems related to their everyday life. The idea of a solution of equations has existed since the time of the ancient Egyptians and Babylonians. Equations with terms raised to a power greater than one can be solved by factoring and, in some specific cases, by the quadratic equation. Solutions for equations with multiple unknown variables are found by using the principles for a system of equations. 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. The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true. Solving second degree and higher equations
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