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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.For example, the stated problem "Find a number which, when added to 3, yields 7" may be written as: 3 ?
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.
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 symbols, a - b, a c = b c, and a - c = b - c are equivalent equations.
Solve And Show Work For Math Problems
Write an equation equivalent to x 3 = 7 by subtracting 3 from each member.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.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.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.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 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.