*For example, the stated problem "Find a number which, when added to 3, yields 7" may be written as: 3 ?*

In this chapter, we will develop certain techniques that help solve problems stated in words.

These techniques involve rewriting problems in the form of symbols.

Since each equation obtained in the process is equivalent to the original equation, -3 is also a solution of 2x 1 = x - 2.

In the above example, we can check the solution by substituting - 3 for x in the original equation 2(-3) 1 = (-3) - 2 -5 = -5 The symmetric property of equality is also helpful in the solution of equations.

Write an equation equivalent to 4x- 2-3x = 4 6 by combining like terms and then by adding 2 to each member.

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Combining like terms yields x - 2 = 10 Adding 2 to each member yields x-2 2 =10 2 x = 12 To solve an equation, we use the addition-subtraction property to transform a given equation to an equivalent equation of the form x = a, from which we can find the solution by inspection. We want to obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other.Equations such as x 3 = 7 are first-degree equations, since the variable has an exponent of 1.The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member. In Section 3.1 we solved some simple first-degree equations by inspection.In this case, we get 2x-2x 9 = 3x- 9-2x 9 9 = x from which the solution 9 is obvious.If we wish, we can write the last equation as x = 9 by the symmetric property of equality.Solution Subtracting 3 from each member yields x 3 - 3 = 7 - 3 or x = 4 Notice that x 3 = 7 and x = 4 are equivalent equations since the solution is the same for both, namely 4.The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.Thus, in the equation x 3 = 7, the left-hand member is x 3 and the right-hand member is 7. However, the solutions of most equations are not immediately evident by inspection.Equations may be true or false, just as word sentences may be true or false. Hence, we need some mathematical "tools" for solving equations.We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result. The first-degree equations that we consider in this chapter have at most one solution. Notice in the equation 3x 3 = x 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection.Determine if the value 3 is a solution of the equation 4x - 2 = 3x 1 Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member. The solutions to many such equations can be determined by inspection. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted.

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