Sometimes one method is better than another, and in some cases, the symmetric property of equality is also helpful. (1) Solution If we first add -3x to each member, we get 2x - 3x = 3x - 9 - 3x -x = -9 where the variable has a negative coefficient.
Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x and 9 to each member of Equation (1).
For example, the stated problem "Find a number which, when added to 3, yields 7" may be written as: 3 ?
= 7, 3 n = 7, 3 x = 1 and so on, where the symbols ? We call such shorthand versions of stated problems equations, or symbolic sentences.
The equation: 3 x = 7 will be false if any number except 4 is substituted for the variable. EQUIVALENT EQUATIONS Equivalent equations are equations that have identical solutions.
The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. Thus, 3x 3 = x 13, 3x = x 10, 2x = 10, and x = 5 are equivalent equations, because 5 is the only solution of each of them.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.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.If we first add -1 to (or subtract 1 from) each member, we get 2x 1- 1 = x - 2- 1 2x = x - 3 If we now add -x to (or subtract x from) each member, we get 2x-x = x - 3 - x x = -3 where the solution -3 is obvious.The solution of the original equation is the number -3; however, the answer is often displayed in the form of the equation x = -3.In symbols, a - b, a c = b c, and a - c = b - c are equivalent equations.Write an equation equivalent to x 3 = 7 by subtracting 3 from each member.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.