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In general, we have the following property, which is sometimes called the division property.
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.
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.
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.