Solve each pair of the equations given below, using substitution method:

Question 1:  x+y=12 x-y=2  

Answer:

x+y=12     … … … … …i)

x-y=2     … … … … …ii)

From ii)

x=y+2

Substituting in i)

y +2+y=12

\Rightarrow 2y=10

\Rightarrow y=5

Therefore x=5+2=7

Hence x=7, y=5

\\

Question 2: 5x+3y=24 3x-y=20

Answer:

5x+3y=24     … … … … …i)

3x-y=20     … … … … …ii)

From ii)

y=3x-20

Substituting in i)

5x+3(3x-20)= 24

\Rightarrow 14x-60=24

\Rightarrow x=6

Therefore y=3\times 6-20= -2

Hence x=6, y= -2

\\

Question 3: x-3y=2 , 2x+7y=30

Answer:

x-3y=2     … … … … …i)

2x+7y=30     … … … … …ii)

From i)

x=3y+2

Substituting in ii)

2(3y+2)+ 7y=30

\Rightarrow 13y+4=30

\Rightarrow y=2

Therefore x=3\times 2+2=8

Hence x=8, y=2

\\

Question 4: x+4y=5 4x+y=50

Answer:

x+4y=5     … … … … …i)

4x+y=50     … … … … …ii)

From i)

x=5-4y

Substituting in ii)

4(5-4y)+ y=50

\Rightarrow 20-25y=50

\Rightarrow y= -2

Therefore x=5-4\times (-2)= 13

Hence x=13, y= -2

\\

Question 5: 2x+3y=6 3x+5y=15

Answer:

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

\Rightarrow  9-4.5y+5y=15

\Rightarrow y=12

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

Hence x= -15, y=12  

\\

Question 6: 5x-7y=9 , 2x+5y=12

Answer:

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

\Rightarrow ( \frac{14}{5} +5 )y- \frac{18}{5} =12

\Rightarrow \frac{39}{5} y= \frac{78}{5}

\Rightarrow y=2

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

Hence x=1, y=2

\\

Solve each pair of the equations given below, using elimination method:

Question 7: x+2y=39 , 2x-3y=1

Answer:

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

\Rightarrow y=11

Substituting in i)

x=39-2\times 11=17

Hence x=17 , y=11

\\

Question 8: 3x-y=5 , 5x-2y=4

Answer:

3x-y=5     … … … … …i)

5x-2y=4     … … … … …ii)

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

6x-2y=10

  (-) \underline{5x-2y=4}

     x=6

\Rightarrow x=6

Substituting in i)

y=3 (6) - 5 =13

Hence x=6 , y=13

\\

Question 9:  14x-3y=54 21x-8y=95

Answer:

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

\Rightarrow y = -4

Substituting in i)

14x=3(-4)+54

x= \frac{42}{14} =3

Hence x=3, y=-4

\\

Question 10: 7x-6y=37 , 5x+4y=43

Answer:

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

\Rightarrow y=2

Substituting in i)

x= \frac{6\times 2+37}{7} =7

Hence x=7, y=2

\\

Question 11: 10x+3y=36 , 15x-14y=17

Answer:

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

\Rightarrow y=2

Substituting in i)

x= \frac{(36-3\times 2)}{10} =3

Hence x=3, y=2

\\

Question 12: 4x+3=14 , 9x-5y=55

Answer:

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

\Rightarrow y=-2

Substituting in i)

x= \frac{((14-3(-2))}{4} =5

Hence x=5, y= -2

\\

Question 13: 4x-3y=11 , 2x-5y= -5

Answer:

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

\\

Solve the following simultaneous equations:

Question 14:  11x-8y=46 2x+7y=-17

Answer:

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

\Rightarrow y= -3

Substituting back in i)

x= \frac{(8(-3)+46)}{11} =2

Hence x=2, y= -3

\\

Question 15:  5x-3y=13 , 3x-2y=5

Answer:

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

\\

Question 16:  5a+4b=22 4a-5b=23

Answer:

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

\Rightarrow b=3

Substituting in i)

a= \frac{(22-4(3))}{5} =2

Hence a=2, b=3

\\

Question 17:  8x+3y=0 , 3x+5(y+3)= -16

Answer:

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

\Rightarrow y= -8

Therefore

x= - \frac{3}{8} \times (-8)= 3

Hence x=3, y= -8

\\

Question 18:  8y-5z=7   ,  3y=4(z-2)

Answer:

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

\Rightarrow z=5

Substituting in i)

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

Hence z=5, y=4

\\

Question 19:  2(a-3)+3(b-5)= 0 , 5(a-1)+4(b-4)= 0

Answer:

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

\Rightarrow b=9

Substituting in iii)

a= \frac{(21-3(9))}{2} = -3

Hence a= -3, b=9

\\

Question 20:  4(3x-y)= 9x+5   ,  3(2x+3y)=13 (x+y-5)

Answer:

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

\Rightarrow x=7

Substitute in iii)

4y=3x-5=21-5

\Rightarrow y= \frac{16}{4} =4

Hence x=7, y=4

\\

Question 21:  \frac{x}{2} - \frac{y}{3} =3 4x-3y=22

Answer:

\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

\Rightarrow y=6

Substituting in ii)

x= \frac{(3\times 6+22)}{4} =10

Hence x=10, y=6

\\

Question 22:  \frac{x}{3} - \frac{y}{4} =0 , \frac{2x}{3} + \frac{3y}{4} =5

Answer:

\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

\Rightarrow y=4

x=6\times \frac{4}{8} =3

Hence x=3, \ y=4

\\

Question 23:  \frac{x}{3} - \frac{5y}{6} =3 , \frac{3x}{4} - \frac{5y}{2} =8

Answer:

\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

\Rightarrow y= -2

Substituting in iii)

x= \frac{(5(-2)+18)}{2} =4

Hence x=4, y= -2

\\

Question 24:  \frac{(3a+5)}{4} = \frac{(2b-1)}{6} , \frac{4a}{3} + \frac{b}{6} = -1

Answer:

\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

\\

Question 25:  23x+31y=7 , 31x+23y=47

Answer:

23x+31y=7     … … … … …i)

31x+23y=47     … … … … …ii)

Add i) and ii)

54x+54y=54

\Rightarrow x+y=1 ... ... ... ... ...iii)

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

23x+31y=7

(-) \underline{23x+23y=23}

8y= -16

\Rightarrow y= -2

x=1-y=1-(-2)=3

Hence x=3, y= -2

\\

Question 26:  97x-78y=59 , 78x-97y=116

Answer:

97x-78y=59     … … … … …i)

78x-97y=116     … … … … …ii)

Add i) & ii)

175x-175y=175

\Rightarrow 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

\\

Question 27:  \frac{1}{x} + \frac{1}{y} =7 , \frac{1}{x} - \frac{1}{y} =1

Answer:

\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

\Rightarrow y= \frac{1}{3}

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

\\

Question 28:  \frac{2}{x} + \frac{10}{y} =3 , \frac{8}{x} - \frac{15}{y} =1

Answer:

\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

\Rightarrow y=5

Substituting

\frac{8}{x} =1+ \frac{15}{y} =4

\Rightarrow x=2

Hence x=2, y=5

\\

Question 29:  2x+ \frac{3}{y} =20 , 4x- \frac{9}{y} =10

Answer:

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

\Rightarrow y= \frac{1}{2}

Substituting

2x=20- \frac{3}{\frac{1}{2}} = 20-6=24

\Rightarrow x=12

Hence x=12, \ and\ y= \frac{1}{2}

\\

Question 30:  \frac{6}{x} -4y=9 , \frac{4}{x} - y=1

Answer:

\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

\Rightarrow x =-2

y= \frac{4}{x} - 1= -2-1= -3

Hence x =-2, y= -3

\\

Question 31:  \frac{3}{2x} - \frac{5}{3y} = \frac{7}{6} , \frac{4}{5x} + \frac{1}{y} =1

Answer:

\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}

\Rightarrow x =1

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

\Rightarrow y=5

Hence x=1, y=5

\\

Question 32:  \frac{3x+2}{2y+3} =\frac{1}{8} , \frac{x+1}{3y-2} =\frac{1}{8}

Answer:

\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

\Rightarrow 9x-2y=-3 ... ... ... ... ...iii)

8x+8=3y-2

\Rightarrow 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

\Rightarrow x=1

Substituting

y= \frac{(9x+3)}{2} = \frac{12}{2} =6

\\

Question 33:  \frac{(2x+1)}{5} - \frac{(3x-y)}{2} =y , \frac{(3x+2)}{2} + \frac{(2-y)}{3} =x-y

Answer:

\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

\Rightarrow 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

\Rightarrow y= -4

Substituting in iii)

x= \frac{2-5(-4)}{11} = \frac{22}{11} =2

Hence x=2, y= -4