Q.1.

x+y=12 ... ... ... ... ...i)

x-y=2 ... ... ... ... ...ii)

From ii)

x=y+2 

Substituting in i)

 y +2+y=12 

 2y=10 

 y=5 

Therefore    x=5+2=7 

Hence    x=7, y=5 

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Q.2.

5x+3y=24 ... ... ... ... ...i)

3x-y=20 ... ... ... ... ...ii)

From ii)

 y=3x-20 

Substituting in i)

 5x+3(3x-20)= 24 

 14x-60=24 

 x=6 

Therefore y=3\times 6-20= -2 

Hence  x=6, y= -2 

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Q.3.

x-3y=2 ... ... ... ... ...i)

2x+7y=30 ... ... ... ... ...ii)

From i)

 x=3y+2 

Substituting in ii)

 2(3y+2)+ 7y=30 

 13y+4=30 

 y=2 

Therefore x=3\times 2+2=8 

Hence x=8, y=2 

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Q.4.

x+4y=5 ... ... ... ... ...i)

4x+y=50 ... ... ... ... ...ii)

m16xFrom i)

 x=5-4y 

Substituting in ii)

 4(5-4y)+ y=50 

 20-25y=50 

 or y= -2 

Therefore x=5-4\times (-2)= 13 

Hence x=13, y= -2 

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Q.5.

2x+3y=6 ... ... ... ... ...i)

3x+5y=15 ... ... ... ... ...ii)

From i)

 x=3-\frac{3}{2}  y 

Substituting in ii)

 3(3- \frac{3}{2}  y )+ 5y=15 

 9-4.5y+5y=15 

 or y=12 

Therefore x=3-\frac{3}{2}\times 12=3-18=-15 

Hence x= -15, y=12    

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Q.6.

5x-7y=9 ... ... ... ... ...i)

2x+5y=12 ... ... ... ... ...ii)

From i)

x=\frac{7}{5} y-\frac{9}{5} 

Substituting in ii)

 2(\frac{7}{5}  y-\frac{9}{5})+ 5y=12 

 (\frac{14}{5}+5 )y-\frac{18}{5}=12 

\frac{39}{5} y=\frac{78}{5} 

 or y=2 

 x=\frac{7}{5}\times 2-\frac{9}{5}=\frac{5}{5}=1 

Hence x=1, y=2 

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Q.7.

x+2y=39 ... ... ... ... ...i)

2x-3y=1 ... ... ... ... ...ii)

Multiply i) by 2 and then subtract ii) from i)

2x+4y=78

      (-)   \underline{2x-3y=1}  

                   7y=77

  or y=11 

 Substituting in i)

x=39-2\times 11=17 m17x

 Hence x=17 , y=11 

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Q.8.

14x-3y=54 ... ... ... ... ...i)

21x-8y=95 ... ... ... ... ...ii)

Multiply i) by 3 and ii) by 2 and subtract ii) from i)

42x-9y=162

(-)  \underline{42x-16y=190}

           7y=-28

Substituting in i)

14x=3(-4)+54 

x=  \frac{42}{14}=3 

Hence x=3, y=-4 

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Q.9.

7x-6y=37 ... ... ... ... ...i)

5x+4y=43 ... ... ... ... ...ii)

Multiply i) by 5 and ii) by 7 and subtract ii) from i)

35x-30y=185

(-)  \underline{35x+28y=301}

      -58y=-116

or y=2

Substituting in i)

x=\frac{6\times 2+37}{7}=7 

Hence x=7, y=2 

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Q.10.

10x+3y=36 ... ... ... ... ...i)

15x-14y=17 ... ... ... ... ...ii)

Multiply i) by 3 and ii) by 2 and subtract ii) from i)

30x+9y=108

(-)   \underline{30x-28y=34}

        37y=74

or y=2 

Substituting in i)

 x=\frac{(36-3\times 2)}{10}=3 

Hence x=3, y=2 

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Q.11.

4x+3=14 ... ... ... ... ...i)

9x-5y=55 ... ... ... ... ...ii)

Multiply i) by 9 and ii) by 4 and subtract ii) from i)

         36x+27y=126 

(-)   \underline{36x-20y=220}

                 47y= -94 

y=-2 

Substituting in i)

 x=   \frac{((14-3(-2)}{4}=5 

Hence x=5, y= -2 

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Q.12.

4x-3y=11 ... ... ... ... ...i)

2x-5y= -5 ... ... ... ... ...ii)

Multiply ii) by 2 and subtract ii) from i)

4x-  3y=11

(-)   \underline{ 4x-10y=-10}

           7y=21

y=3 

Substituting in i)

x=  \frac{(11+3(3))}{4}=5 

Hence x=5, y=3 

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Q.13.

11x-8y=46 ... ... ... ... ...i)

2x+7y=-17 ... ... ... ... ...ii)

Multiply i) by 2 and ii) by 11 and subtract ii) from i)

22x-16y=92

(-) \underline{22x+77y=187}

      -93y=279

or y= -3 

Substituting back in i)

x=  \frac{(8(-3)+46)}{11}=2 

Hence x=2, y= -3 

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Q.14.

5x-3y=13 ... ... ... ... ...i)

3x-2y=5 ... ... ... ... ...ii)

Multiply i) by 3 and ii) by 5 and subtract ii) from i)

15x-9y=39

(-)     \underline{15x-10y=25}

y=14

Substituting back in i)

x=  \frac{(3(14)+13)}{5}=11 

Hence x=11, y=14 

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Q.15.

5a+4b=22 ... ... ... ... ...i)

 4a-5b=23 ... ... ... ... ...ii)

Multiply i) by 4 and ii) by 5 and subtract ii) from i)

20a+16b=88

(-)  \underline{20a+25b=115}

                 -9b= -27

or b=3 

Substituting in i)m18x

a=  \frac{(22-4(3))}{5}=2 

Hence  a=2, b=3 

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Q.16.

8x+3y=0 ... ... ... ... ...i)

3x+5(y+3)= -16 ... ... ... ... ...ii)

Simplifying ii)

 3x+5y= -31 ... ... ... ... ...iii)

From i)

x=-\frac{3}{8} y 

Substituting in iii)

3(-\frac{3}{8} y)+5y= -31 

(5-\frac{9}{8})y=-31 

\frac{31}{8}y= -31 

or y= -8 

Therefore

x= -\frac{3}{8}\times (-8)= 3 

Hence x=3,  y= -8 

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Q.17.

8y-5z=7 ... ... ... ... ...i)

3y=4(z-2) ... ... ... ... ...ii)

Simplify ii)

3x-4z= -8 ... ... ... ... ...iii)

Multiply i) by 3 and iii) by 8 and subtract iii) from i)

24y-15z=21

(-)      \underline{y-32z=-64}

                17z=85

or z=5 

Substituting in i)
y=  \frac{(5(5)+7)}{8}=4 

Hence z=5, y=4 

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Q.18.

2(a-3)+3(b-5)= 0 ... ... ... ... ...i)

5(a-1)+4(b-4)= 0 ... ... ... ... ...ii)

Simplify i) and ii)

 2a+3b=21 ... ... ... ... ...iii)

 5a+4b=21 ... ... ... ... ...iv)

Multiplying iii) by 5 and iv) by 2 and subtract iv) from iii)

10a+15b=105

(-)       \underline{10a+8b=42}

7b=63

or b=9 

Substituting in iii)

a=  \frac{(21-3(9))}{2}= -3 

Hence a= -3, b=9 

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Q.19.

4(3x-y)= 9x+5 ... ... ... ... ...i)

3(2x+3y)=13 (x+y-5) ... ... ... ... ...ii)

Simplifying i) and ii)

 12x-4y=9x+5 

 3x-4y=5 ... ... ... ... ...iii)

 6x+9y=13x+13y-65 

 7x+4y=65 ... ... ... ... ...iv)

Add iii) and  iv)

 10x=70 

 x=7 

Substitute in iii)

 4y=3x-5=21-5 

y=\frac{16}{4}=4 

Hence x=7,  y=4 

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Q.20.

\frac{x}{2}-\frac{y}{3}=3 ... ... ... ... ...i)

4x-3y=22 ... ... ... ... ...ii)

Simplifying i) multiply by 6

3x-2y=18 ... ... ... ... ...iii)

Multiply ii) by 3 and iii) by 4 and subtract iii) from ii)

12x-9y=66

(-)      \underline{12x-8y=72}

-y= -6

or y=6 

Substituting in ii)

x=\frac{(3\times 6+22)}{4}=10 

Hence  x=10, y=6 

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Q.21.

\frac{x}{3}-\frac{y}{4}=0 ... ... ... ... ...i)

\frac{2x}{3}+\frac{3y}{4}=5 ... ... ... ... ...ii)

Simplify i) and ii) by multiplying i) by 12 and ii) also by 12

4x-3y=0 ... ... ... ... ...iii)

8x+9y=60 ... ... ... ... ...iv)

Multiply iii) by 2 and subtract  iv) from iii)

8x-6y=0

(-)      \underline{8x+9y=60}

-15y= -60

or y=4 

x=6\times \frac{4}{8}=3 

Hence x=3, \ y=4 

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Q.22.

\frac{x}{3}-\frac{5y}{6}=3 ... ... ... ... ...i)

\frac{3x}{4}-\frac{5y}{2}=8 ... ... ... ... ...ii)

Simplify i) and ii) by multiplying i) by 6 and ii) by 4

 2x-5y=18 ... ... ... ... ...iii)

 3x-10=32 ... ... ... ... ...iv)

Now multiply iii) by 3 and iv) by 2 and subtract iv) from iii)

 6x-15y=54

 (-)     \underline{6x-20y=64}

 5y= -10

or y= -2 

Substituting in iii)

x=\frac{(5(-2)+18)}{2}=4 

Hence x=4, y= -2 

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Q.23.

\frac{(3a+5)}{4}=\frac{(2b-1)}{6} ... ... ... ... ...i)

\frac{4a}{3}+\frac{b}{6}= -1 ... ... ... ... ...ii)

Simplify i) and ii), multiply i) by 12, and ii) by 6

 3(3a+5)= 2(2b-1) 

 9a+15=4b-2 ... ... ... ... ...iii)

 9a-4b= -17 

 8a+b= -6 ... ... ... ... ...iv)

Multiply iv) by 4 and add iii) and iv)

 9a-4b=-17

 (+)     \underline{32a+4b=-24}

 41a    =     -41

a= -1 

Substituting in iv)

 b= -8(-1)-6=2 

Hence a= -1, b=2 

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Q.24.

23x+31y=7 ... ... ... ... ...i)

 31x+23y=47 ... ... ... ... ...ii)

Add i) and ii)

 54x+54y=54 

 or x+y=1 ... ... ... ... ...iii)

Now multiply iii) by 23 and subtract iii) from i)

23x+31y=7

(-)    \underline{23x+23y=23}

8y= -16

or y= -2 

x=1-y=1-(-2)=3 

Hence x=3, y= -2 

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Q.25.

97x-78y=59 ... ... ... ... ...i)

78x-97y=116 ... ... ... ... ...ii)

Add i) & ii)

 175x-175y=175 

 or x-y=1 ... ... ... ... ...iii)

Now multiply iii) by 97 and subtract iii) from i)

97x-78y=59

(-)     \underline{97x-97y=97}

19y= -38

y= -2 

Substitute x=y+1= -2+1=-1 

Hence x= -1, y= -2 

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Q.26.

\frac{1}{x}+\frac{1}{y}=7 ... ... ... ... ...i)

\frac{1}{x}-\frac{1}{y}=1 ... ... ... ... ...ii)

Add i) & ii)

\frac{2}{x}=8 

x=\frac{1}{4}

Substituting in i)

\frac{1}{y}=7-\frac{1}{\frac{1}{4}}=7-4=3 

or y=\frac{1}{3}

Hence,   x=\frac{1}{4}  \ and\  y=\frac{1}{3}

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Q.27.

\frac{2}{x}+\frac{10}{y}=3 ... ... ... ... ...i)

\frac{8}{x}-\frac{15}{y}=1 ... ... ... ... ...ii)

Multiply i) by 4 and subtract ii) from i)

\frac{8}{x}+\frac{40}{y}=12

(-)     \underline{\frac{8}{x}-\frac{15}{y}=1}

\frac{55}{y}=11

or  y=5 

Substituting

\frac{8}{x}=1+\frac{15}{y}=4 

or x=2 

Hence x=2, y=5 

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Q.28.

2x+\frac{3}{y}=20 ... ... ... ... ...i)

4x-\frac{9}{y}=10 ... ... ... ... ...ii)

Multiply i) by 2 & then subtract ii) from i)

4x+\frac{6}{y}=40

(-)     \underline{4x-\frac{9}{y}=10}

\frac{15}{y}=30

or y=\frac{1}{2}

Substituting

2x=20-\frac{3}{\frac{1}{2}}= 20-6=24 

or x=12 

Hence x=12, \ and\  y=\frac{1}{2}

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Q.29.

\frac{6}{x}-4y=9 ... ... ... ... ...i)

\frac{4}{x}- y=1 ... ... ... ... ...ii)

Multiply ii) by 4 & then subtract ii) from i)

\frac{6}{x}- 4y=9

(-)     \underline{\frac{16}{x}- 4y=4}

-\frac{10}{x}=5 

or x =-2 

y=\frac{4}{x}- 1= -2-1= -3 

Hence x =-2,   y= -3 

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Q.30.

\frac{3}{2x}-\frac{5}{3y}=\frac{7}{6} ... ... ... ... ...i)

\frac{4}{5x}+\frac{1}{y}=1 ... ... ... ... ...ii)

Multiply ii) by \frac{5}{3} and add i) & ii)

\frac{3}{2x}-\frac{5}{3y}=\frac{7}{6}

(+)    \underline{\frac{20}{15x}+\frac{5}{3y}=\frac{5}{3} }

(\frac{3}{2}+\frac{4}{3})(\frac{1}{x})=(\frac{7}{6}+\frac{5}{3}) )

\frac{17}{6}(\frac{1}{x})=\frac{51}{18}

x =1 m19x

Substituting  \frac{1}{y}=1-(\frac{4}{5})=1/5 

 y=5 

Hence x=1, y=5 

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Q.31.

\frac{3x+2}{2y+3}=\frac{1}{8} ... ... ... ... ...i)

\frac{x+1}{3y-2}=\frac{1}{8} ... ... ... ... ...ii)

Simplify i) and ii)

 9x+6=2y+3 

 9x-2y=-3 ... ... ... ... ...iii)

 8x+8=3y-2 

 8x-3y= -10 ... ... ... ... ...iv)

Multiply iii) by 3 and iv) by 2 and subtract iv) from iii)

 27x-6y=-9

 (-)     \underline{16x-6y= -20}

 11x-11

x=1 

Substituting

y=\frac{(9x+3)}{2}=\frac{12}{2}=6 

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Q.32.

\frac{(2x+1)}{5}-\frac{(3x-y)}{2}=y ... ... ... ... ...i)

\frac{(3x+2)}{2}+\frac{(2-y)}{3}=x-y ... ... ... ... ...ii)

Simplify i) and ii)

 2(2x+1)-5(3x-y)=10y 

 4x+2-15x+5y=10y 

 -11x-5y= -2 

 or 11x+5y=2 ... ... ... ... ...iii)

 3(3x+2)+2(2-y)=6(x-y) 

 9x+6+4-2y=6x-6y 

 3x+4y=-10 ... ... ... ... ...iv)

Multiply ii) by 3 and iv) by 11 and then subtract iv) from ii)

 33x+15y=6

 (-)      \underline{33x+4y= -110}

 -29y=118

or y= -4 

Substituting in iii)

x=\frac{2-5(-4)}{11}=\frac{22}{11}=2 

Hence x=2,  y= -4 

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