which statement is correct about the values of x and y in the following equation?

Question

7x+xy= xy+21

1. the equation is true for all ordered pairs
2. there are no x and y pairs for which the equation is true.
3. For each value of x, there is one and only one value of y that makes the equation true
4. for each value of y, there is one and only on value of x that makes the equation true

Answer 1

tricky ... let's see ...

we notice that if we subtract xy from both sides we get

7x + xy - xy = xy - xy + 21

then

7x = 21

and

x = 21/7 = 3

so there is only one value of x that satisfies the equation

x = 3

Going back to the original equation we see that any value of y will satisfy the original equation

we can see this by rearranging things:

7x + xy - xy = 21

here, I have performed the subtraction of xy on the right side as above, but have left the left side undone

(so we don't lose the presence of y)

Note that the above can also be written as

7x + (x - x)y = 21

or

7x + 0y =21

now, since anything times zero equals zero,

y may be any number.

Let's summarize:

1) x = 3

2) y = anything

looking back at the original question;

1) the equation is true for all ordered pairs

FALSE (only one x works, not all x)

2) there are no x and y pairs for which the equation is true.

FALSE (x=3, y=anything) makes it true, i.e. (3,1)

3) For each value of x, there is one and only one value of y that makes the equation true,

FALSE for each value of x, for the one value of x, x=3, y can be any number, which is an infinite number, not one

4) for each value of y, there is one and only on value of x that makes the equation true.

TRUE!! for all the infinite values of y you may pick, there is one and only one x you may pick, x = 3

ANSWER: STATEMENT 4 is CORRECT (TRUE)

Answer 2

which statement is correct about the values of x and y in the following equation?

7x+xy= xy+21

Subtract xy from both sides and this is: 7x = 21

x = 3

the equation is true for all ordered pairs

To show that this statement is FALSE, all you need to do is find one example (a counterexample) where the statement is FALSE. How about the ordered pair (0,0). 0 ≠ 21 [x must always be 3]

there are no x and y pairs for which the equation is true.

To show that this statement is FALSE, all you need to do is find one example where the statement is FALSE (note: where the equation is TRUE).

Let's make it easy and use (3,7) 7*3 + 3*7 = 3*7 + 21

42 = 42

For each value of x, there is one and only one value of y that makes the equation true

This is the "vertical line test." It FAILS. Here are two points with the same value of x: (3,7) and (3,10)

for each value of y, there is one and only on value of x that makes the equation true

This is the horizontal line test." FOR EVERY VALUE OF Y, x must equal 3.

Answer 3

7x + xy = xy + 21

Subtract xy from both sides to get

7x = 21

Divide both sides by 7 to get x=3.

Any value of y when x=3, the one and only one value of x, will make the equation true.

The answer is #4.