*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.The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.*

*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.

Consider the equation 3x = 12 The solution to this equation is 4.

Also, note that if we divide each member of the equation by 3, we obtain the equations whose solution is also 4.

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 and a·c = b·c (c ≠ 0) are equivalent equations.

Write an equivalent equation to by multiplying each member by 6.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.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.In general, we have the following property, which is sometimes called the multiplication property.If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to the original 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.This property states If a = b then b = a This enables us to interchange the members of an equation whenever we please without having to be concerned with any changes of sign.Thus, If 4 = x 2 then x 2 = 4 If x 3 = 2x - 5 then 2x - 5 = x 3 If d = rt then rt = d There may be several different ways to apply the addition property above.

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