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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.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.
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.
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.
Write an equation equivalent to 4x- 2-3x = 4 6 by combining like terms and then by adding 2 to each member.
Combining like terms yields x - 2 = 10 Adding 2 to each member yields x-2 2 =10 2 x = 12 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. 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.
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.
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.