*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.*

Solution Dividing both members by -4 yields In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1. We first combine like terms to get 5y = 20 Then, dividing each member by 5, we obtain In the next example, we use the addition-subtraction property and the division property to solve an equation. Solution First, we add -x and -7 to each member to get 4x 7 - x - 7 = x - 2 - x - 1 Next, combining like terms yields 3x = -9 Last, we divide each member by 3 to obtain Consider the equation The solution to this equation is 12.

Also, note that if we multiply each member of the equation by 4, we obtain the equations whose solution is also 12.

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.

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.In general, we have the following property, which is sometimes called the division property.If both members of an equation are divided by the same (nonzero) quantity, the resulting equation is equivalent to the original equation. Write an equation equivalent to -4x = 12 by dividing each member by -4.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.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).The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.If the same quantity is added to or subtracted from both members of an equation, the resulting equation is equivalent to the original equation.

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