in this lesson we’ll examine a

systematic technique for drawing conclusions about expressions involving

even and odd numbers to demonstrate this technique we’ll examine a question we

saw earlier now in this question we must determine whether X is even and the two

statements involve expressions that evaluate to be odd numbers let’s begin

with statement 1 it tells us that XY plus y is odd now this expression

involves two variables x and y and each variable can be either odd or even so

there are several different cases to consider X and y can both be odd or X

can be even and y can be odd and so on now each of these cases will cause the

expression XY plus y to evaluate to be either odd or even to keep track of the

different cases we’ll use a table and for each case we’ll determine whether

the expression evaluates to be odd or even so what are the possible cases here

well x and y can both be even X can be even and y can be odd X can be odd and Y

can be even or both variables can be odd so there are four different cases to

consider now let’s see what effect each case has on our expression XY plus y so

for the first case we’ll replace the XS and the Y’s with E to show that they are

both even to evaluate this we’ll apply our rules regarding even and odd numbers

so in the first case we have an even number times an even number which is

even and to this we add an even number which will result in an even number so

for the first case where X and y are both even the expression XY plus y

evaluates to be even for the next case will plug in YZ and those for the even

and odd numbers when we apply our rules we see that the expression evaluates to

be an odd number we’ll follow these same steps for the other two cases as well so

we now know the outcomes in each of the four cases statement 1 tells us that the

expression evaluates to be an odd number so when we examine our table we see that

there is only one case where the expression evaluates to be an odd number

in this particular case X is even and Y is odd so since we can now be certain

that X is even we can answer the target question which means statement one is

sufficient now some students may find the cumbersome to plug YZ and O’s into

the expression and then apply these rules for each case so another strategy

is to plug in actual numbers into our expression for example if x and y are

both even we can replace both variables with twos when we do this our expression

evaluates to be 6 which is even now it doesn’t matter which odd numbers and

which even numbers you plug in however it’s a good idea to use small numbers to

make our calculations easier so for odd numbers plugging in a one will make

calculations easy and for even numbers 2 is a good number to use now using a zero

for even numbers can make calculations even easier however the only drawback is

that a zero might get confused with an O which represents odd numbers the

important point here is to use small values when plugging in numbers ok for

the next case we’ll plug in a 2 for the even number and a 1 for the odd number

when we do this our expression evaluates to be 3 which is odd and we can continue

this method with the other two cases and when we do so we see that our results in

this table are the same as they were in the first table only one case results in

the expression evaluating to be an odd number so once again we can be certain

that X is even which means statement one is sufficient okay now want the

statement to we’ll use the same strategy to examine the four possible cases so

for the first case we’ll plug in YZ to represent even numbers and then apply

our rules to see that the expression evaluates to be an even number following

the same steps we can test the other three cases as well alternatively we can

plug in even and odd into our expression and then evaluate

them to get identical results we have now tested all possible cases

now statement 2 tells us that the expression evaluates to be an odd number

when we examine either of our tables we see that there are two possible cases

where the expression evaluates to be an odd number in one case X is even and in

the other case X is odd so since X can be either even or odd statement two is

not sufficient which means the correct answer here is a so we now have a

systematic technique for drawing conclusions about expressions involving

even and odd numbers so when you encounter a question involving even and

odd numbers consider creating a table to test the various cases if you use a

table you can let either those represent even and odd numbers and then apply the

following rules or you can plug even and odd numbers into the expression and then

evaluate finally once you have tested each case you can draw conclusions based

on the various outcomes