Explanation
Given equations are x2+y5=5 and 2x+5y=56
Let x2=u and y5=v
u+v=5 .....(i)
1u+1v=56 .....(ii)
⇒u+vuv=56⇒5uv=56⇒uv=6 .........(iii)
From (i) v=5−u
(5−u)u=6⇒5u−u2=6⇒u2−5u+6=0⇒u2−2u−3u+6=0⇒u(u−2)−3(u−2)=0⇒(u−3)(u−2)=0⇒u=2,3
But x2=u
⇒x=2u⇒x=4,6
Also, y5=v
⇒y=5v⇒y=15,10
let(1x+y)=uand(1x−y)=vthen22u+165v=5→(1.)and55+45v=14→(2.)multiplyequation(1.)by3,weget66u+45v=15→(1.)+55u+45v=14→(2.)uponsubtractionof(1.)with(2.),weget11u+0=1∴u=(111)orx+y=11→(3.)andthenv=(5−22u15)=(5−215)=(15)∴x−y=5→(4.)add(3.)and(4.)equationweget2x=16∴x=8andy=11−x=11−8=3
x10+y5−1=0
x10+y5=1
x+2y10=1
x+2y=10.....................(1)
now consider,x8+y6=15
6x+8y48=15
3x+4y=24×15=360.................................(2)
now multiply (1)by 3
3x+6y=30 ........................(3)
subtracting (2) and (3)
2y=−330
y=−165
put in(1)
x+2(−165)=10
x−330=10
x=340
now, y=λx+5
⟹−165=λ(340)+5
−170=λ(340)
λ=−170340=−12
3x+4y=10 .....(1)
2x−2y=2 .....(2)
equation (2) can be written as
⇒x−y=1
⇒x=y+1 .....(3)
substitute x value in equation (2)
⇒3x+4y=10
⇒3(y+1)+4y=10
⇒3y+3+4y=10
⇒7y=7
⇒y=1
then, substitute y value in equation (3)
⇒x=y+1=1+1=2
∴x=2,y=1
Plot each line by taking two points on them.2x−3y=1x=0⟹y=−13 and y=0⟹x=12Join the two points (0,−13) and (12,0) by drawing line.
4x−3y+1=0
x=0⟹y=−13and y=0⟹x=14
Join the two points (0,−13) and (14,0) by drawing line.
From the graph, the point of intersection of both lines is (−1,−1) and hence solution of given system of linear equation is x=−1,y=−1
Step -1: Form the required equations.
Let the number of girls in class A be x.
Let the number of girls in class B be y.
Let the number of boys in class A be 35−x.
Let the number of boys in class B be 35−y.
If seven girls are shifted from class A to B,
Number of girls in class A would become x−7.
Number of girls in class B would become y+7.
But according to the question,
y=x−7
⇒x−y=7…(i)
If four girls are shifted from class B to A,
Number of girls in class A would become x+4.
Number of girls in class B would become y−4.
x+4=2y
⇒x−2y=−4…(ii)
Step -2: Solve the above formed two equations simultaneously.
On subtracting equation (ii) from (i), we get
−y+2y=7+4
⇒y=11
On putting the value of y in equation (i), we get
x−11=7
⇒x=18
Step -3: Find the number of boys in class A and B.
Number of boys in class A=35−18
=17
Number of boys in class B=35−11
=24
Therefore, option D is correct.
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