If we now add -x to (or subtract x from) each member, we get If we first add -1 to (or subtract 1 from) each member, we get 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. 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. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.Д®xample 2 Write an equation equivalent toД«y combining like terms and then by adding 2 to each member. Notice that x + 3 = 7 and x = 4 are equivalent equations since the solution is the same for both, namely 4. Solution Subtracting 3 from each member yields Of an equation, the resulting equation is equivalent to the originalД®xample 1 Write an equation equivalent to If the same quantity is added to or subtracted from both members The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted. 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. Thus,ДЈx + 3 = x + 13, 3x = x + 10, 2x = 10, and x = 5ДЄre equivalent equations, because 5 is the only solution of each of them. Hence, we need some mathematical 'tools' for solving equations.Д®quivalent equations are equations that have identical solutions. However, the solutions of most equations are not immediately evident by inspection. In Section 3.1 we solved some simple first-degree equations by inspection. SOLVING EQUATIONS USING ADDITION AND SUBTRACTION PROPERTIES The solutions to many such equations can be determined by inspection.Д®xample 2 Find the solution of each equation by inspection. The first-degree equations that we consider in this chapter have at most one solution. Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member. 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.Д®xample 1 Determine if the value 3 is a solution of the equation The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. Will be false if any number except 4 is substituted for the variable. SOLVING EQUATIONSД®quations may be true or false, just as word sentences may be true or false. Thus, in the equation x + 3 = 7, the left-hand member is x + 3 and the right-hand member is 7. 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. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. We call such shorthand versions of stated problems equations, or symbolic sentences. 'Find a number which, when added to 3, yields 7'ДЄnd so on, where the symbols ?, n, and x represent the number we want to find. These techniques involve rewriting problems in the form of symbols. In this chapter, we will develop certain techniques that help solve problems stated in words.

5 x 1 answer