A System of Nonlinear Equations

Okay.

First we add the two equations to find out $(x+y)$ .

$x(x+y) + y(x+y) = 25$

$\Rightarrow (x+y)^2 = 5^2$

Now to get $xy$ we multiply the two equations :

$xy \cdot (x+y)^2 = 9 \cdot 16$

$\Rightarrow xy = \dfrac{144}{25}$

Which is of the form $\dfrac{a}{b}$

Now since , $a$ and $b$ are coprime ,

$a + b = 144 + 25 = \boxed{169}$

Therefore , $169$ is the correct answer .

Cheers!

Adding the equations after distribution,

$x^2+xy=9$ plus

$+xy+y^2=16$ equals

$x^2+2xy+y^2=25$

Recognizing the perfect squares,

$(x+y)^2=5^2$ .

Taking square roots,

$x+y=5$ .

Substituting,

$x(5)=9$ and $y(5)=16$ .

Solving for variables,

$x=\frac{9}{5}$ and $y=\frac{16}{5}$ .

Multiplying for the answer $xy$ ,

$\frac{144}{25}$ means $a+b=169$ .

X(X+Y)=9.....EQN.1 Y(X+y)=16....EQN.2 Dividing 1 by 2. We get:X=9Y Now substracting 1 and 2 we get: X(X+Y)-Y (X+Y)=-7 Y.Y-X.X =7 By Substituting: we get: Y=16/5 and X=9/5 Then XY=144/25 X+Y=144+25=169

If we let

x ( x + y ) = 9 be (eq 1)

and

y ( x + y ) = 16 be (eq 2)

then (1) + (2):

x ( x + y ) + y ( x + y ) = 9 + 16 = 25

by factoring:

$( x + y ) \times ( x + y )$ = $5 \times 5$

or

$( x + y )^{2}$ = $5^{2}$

taking the square root of both sides, we will get

x + y = 5 (eq 3) or x + y = -5 (eq 4)

Substitute (eq 3) and (eq 4) to (eq 1) and (eq 2) and simlifying, we can get

x = $\frac{9}{5}$ or $\frac{-9}{5}$

y = $\frac{16}{5}$ or $\frac{-16}{5}$

Therefore

xy = $\frac{9}{5} \times \frac{16}{5}$ or $\frac{-9}{5} \times \frac{-16}{5}$

which both yields

xy = $\frac{144}{25}$

Therefore, a = 144 and b = 25 and a + b = 169

$x^{2}+xy=9$

$y^{2}+xy=16$

Adding the two equations give us this:

$x^{2}+2xy+y^{2}=25$

$(x+y)^{2}=25$

$x+y=5$

Substituting into the original, we get:

$5x=9$

$x=\frac{9}{5}$

$5y=16$

$y=\frac{16}{5}$

Multiplying it together, we get $\frac{16}{5}\times\frac{9}{5}=\frac{144}{25}$

$144+25=\boxed{169}$

we note that x+y=a

so ax=9 and ay=19

ax+ay=25 then we can simplified them to a^2=25.

we multiply the first term and second term which we will got xya^2=144

OK now is the last step to find the answer by replace a^2 with 25

so a+b =169

sum all of the equation and you'll get x^2+2xy+y^2=25

(x+y)^2=25 >>> x+y=5

therefore 5x=9 >>>x=9/5

5y=16 >>>> y=16/5

xy=9/5*16/5=144/25=a/b

a+b= 144+25=169

On adding both equations, we get -

$x^2 + 2xy + y^2 = 25 \Rightarrow x+y=5$

On subtracting $eqn(ii)$ from $eqn(i)$ , we get-

$y^2-x^2=7 \Rightarrow (y+x)(y-x)=7 \Rightarrow y-x=\frac{7}{5}$

This gives $x=\frac{9}{5}$ and $y=\frac{16}{5}$ . Hence $xy=\frac{144}{25}=\frac{a}{b}$

Therefore, $a+b=169$

Dividing the two equations, we get $$\frac{x}{y}=\frac{9}{16} \Rightarrow \frac{y}{x}=\frac{16}{9}.$$ Expanding the equations and adding, we get $$x^2+y^2+2xy=25.$$ Because $x^2+y^2$ can be expressed as $xy(\frac{x}{y}+\frac{x}{y})$ , we can rewrite our summed equation as $$xy(\frac{x}{y}+\frac{x}{y})+2xy=25.$$ But $xy(\frac{x}{y}+\frac{x}{y})=xy(\frac{9}{16}+\frac{16}{9})=\frac{337xy}{144}$ . Now our summed equation becomes $$\frac{337xy}{144}+2xy=25 \Rightarrow \frac{625xy}{144}=25 \Rightarrow xy=\frac{144}{25}=\frac{a}{b}.$$ Therefore, $a+b=144+25=\boxed{169}.$

x(x+y)=9 ......eqn 1

y(x+y)=16 ......eqn 2

eqn 1 + eqn 2

we get:

x^2 + 2 x*y + y^2 = 9+16

(x+y)^2=25

(x + y)=sqrt(25)

x+y=5 .....eqn 3

sub eqn 3 in eqn 1

we get:

x(5)=9

x=9/5

sub eqn 3 in eqn 2

we get:

y(5)=16

y=16/5

to find: x*y

x y=(9/5) (16/5)

x*y=144/25 ......eqn 4

given that x*y=a/b .....eqn 5

comparing eqn 4 and eqn 5

we get: a=144 , b=25

(a+b)=144+25

(a+b)=169

ANS 169

From the first equation we get $(x+y)=\frac{9}{x}$ putting this in second equation we get $y=\frac{16x}{9}$ putting this again in the second equation $x=+-\frac{9}{5}$ and $y=+-\frac{16}{5}$ .As $x$ and $y$ should be of the same sign to satisfy the equations $xy=\frac{144}{25}$

x(x+y) = 9 y(x+y) = 16

x^2 + xy = 9 y^2 + xy = 16

adding the two eq.

(x+y)^2 = 25 x + y = 5

substituting to the first eqs.

x(5) = 9 x = 9/5

y(5) = 16 y = 16/5

Hence, xy = (16/5)(9/5) = 144/25

therefore a+b = 169