Cancelation

Pronunciation: /ˈkæn səl æ ʃən/ ?

To cancel is to simplify an expression using sums totaling 0 and common factors.[1] Cancelation is the process of canceling.

Example 1: Simplify x+3=5
StepExpressionExplanation
1x+3=5In order to get x by itself on the left side of the equation, the 3 will be canceled.
2x+3-3=5-3 so 3 is subtracted from both sides.The 3 will be canceled. Create a zero sum by subtracting 3 from both sides. This uses the subtractive property of equality.
3x+3-3=5-3 where 3-3 is the zero sum and 5-3=2.Group the numeric expressions. 3-3 is the zero sum on the left, and 5-3=2 on the right. The associative property of addition allows one to change the order of the terms, if necessary.
4x+0=2 where x+0=0.Simplify the numeric expressions using the substitution property of equality. Use the additive property of zero to change x+0 to x.
5x=2.The solution to the equation is x=2.
Table 1

Example 2: Simplify x+y=3y
StepExpressionExplanation
1x+y=3yIn order to get x by itself on the left side of the equation, the y on the left side of the equation will be canceled.
2x+y-y=3y-y so y is subtracted from both sides.The y will be canceled. Create a zero sum by subtracting y from both sides. This uses the subtractive property of equality.
3x+y-y=3y-y where y-y is the zero sum and 3y-y=2y.Group the numeric expressions. y-y is the zero sum on the left, and 3y-y=2y on the right. The associative property of addition allows one to change the order of the terms, if necessary.
4x+0=2y where x+0=0.Simplify the numeric expressions using the substitution property of equality. Use the additive property of zero to change x+0 to x.
5x=2y.The solution set of the equation is x=2y.
Table 2

Example 3: Simplify 2x/6
StepExpressionExplanation
12x/6The fraction 4x/6 will be simplified using cancellation.
22x/(2*3)Factor the numerator and denominator of the fraction. 4x=2·2·x and 6 = 2·3.
32x/6=(2/2)*(x/3)Find the common factors of the numerator and denominator. The common factor is 2. Use the associative property of multiplication to group the common factors together.
4(2/2)*(x/3)=1*(x/3)Since anything divided by itself is 1, use the substitution property of equality to substitute 1 for 2/2.
51*(x/3)=x/3Use the multiplicative property of 1 to eliminate the 1 from the expression.
62x/6=x/3The fraction 4x/6 can be simplified to 2x/3.
Table 3

Example 4: Simplify 6xy/10x
StepExpressionExplanation
16xy/10xThe fraction 6xy/10x will be simplified using cancellation.
26xy/10x=(2*3*x*y)/(2*5*x)Factor the numerator and denominator of the fraction. 6xy=2·3·x·y and 10x=2·5·x.
3(2*3*x*y)/(2*5*x)=(2*x)/(2*x)*(3*y)/(5)Find the common factors of the numerator and denominator. The common factor is 2x. Use the associative property of multiplication to group the common factors together.
4(2*x)/(2*x)*(3*y)/(5)=1*(3*y)/(5)Since anything divided by itself is 1, use the substitution property of equality to substitute 1 for 2x/2x.
51*(3*y)/(5)=3y/5Use the multiplicative property of 1 to eliminate the 1 from the expression.
66xy/10x=3y/5,x!=0The fraction 6xy/10x can be simplified to 3y/5, assuming x is not 0.
Table 3

Cite this article as:


Cancelation. 2010-01-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cancel.html.

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Revision History


2010-01-04: Added "References" (McAdams, David.)
2008-12-24: Expanded examples and used images for equations (McAdams, David.)
2008-04-22: Initial version (McAdams, David.)

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