Question

The difference of two numbers is 7 and theirproduct is 170. What are the numbers ?​

Answer 1

ANSWER :

Two numbers are -10 and -17 or 17 and 10.

EXPLANATION :

GIVEN:-

• The difference of two numbers is 7.
• Their product is 170.

TO FIND:-

• The two numbers.

SOLUTION :-

Let the two numbers be x and y.

✪Difference of two numbers=7

✪Product of two numbers = 170

According to the question,

$\sf{x-y=7............(i)}$

$\sf{xy=170.............(ii)}$

† Taking eq (i) †

$\sf{x-y=7}$

$\implies\sf{x=7+y.............(i)}$

★ Now take the eq(ii) and put the value of x=7+y from eq(i) ★

$\sf{xy=170}$

$\implies\sf{(7+y)y=170}$

$\implies\sf{7y+y^2=170}$

$\implies\sf{y^2+7y-170=0}$

$\implies\sf{y^2+(17-10)y-170=0}$

$\implies\sf{y^2+17y-10y-170=0}$

$\implies\sf{y(y+17)-10(y+17)=0}$

$\implies\sf{(y+17)(y-10)=0}$

Now,

$\sf{y+17=0}$

$\implies\sf{y=-17}$

And,

$\sf{y-10=0}$

$\implies\sf{y=10}$

✪ Now putting the value of y=-17 and y=10 in eq(i)✪

$\sf{x-y=7}$

$\implies\sf{x-(-17)=7}$

$\implies\sf{x+17=7}$

$\implies\sf{x=7-17}$

$\implies\sf{x=-10}$

And,

$\sf{x-y=7}$

$\implies\sf{x-10=7}$

$\implies\sf{x=7+10}$

$\implies\sf{x=17}$

Therefore,

x = -10 or 17

y = -17 or 10

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Answer 2

GIVEN:

★The difference of two numbers is 7.

★The product of those two numbers is 170.

TO FIND:

★The two numbers.

CONCEPT USED:

★Here we are given to simultaneous equation which we will be solving by substitution method.

★By doing so, we would get a quadratic equation, whose roots will led to finding our answers.

ANSWER:

Let us take the two numbers be x and y.

First case:

$.°.x-y=7$ ................. (1)

Second case:

=>$x\times y=170$

=>$.°.xy =170$

=>$x =\dfrac{170}{y}$...............(2)

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Using equation 2 in equation 1,we have;

=>$\dfrac{170}{y}-y=7$

=>$\dfrac{170-y^{2}}{y}=7$

=>$170-7y^{2}=7$

=>$y^{2}+7y-170=0$

On splitting middle term,

=>$y^{2}+17y-10y-170=0$

=>$y(y+17) -10(y+17) =0$

=>$(y+17) (y-10) =0$

Hence x = 10,-17.

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We have now two cases

Case1

when y = 10

Then,

=>$x - y =7$

=>$x -10=7$

.°.$x = 17$

______________________________________

Case 2

when y = -17.

Then,

=>$x -y =7$

=>$x+17=7$

=>$x = 7-17$

.°.$x = -10.$

______________________________________

$\huge\orange{\boxed{x=-10\:and\:y=-17}}$

Or

$\huge\purple{\boxed{x =17\:and\:y=10}}$