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Worksheets for Linear equations in two variable




Given below are the linear equations in two variables class 10 worksheet with solutions


Question 1.
Solve these linear equation in two variable ( x and y)
a. 37x+41y=70
41x+37y=86

b. 99x+101y=499
101x+99y=501

c. 23x29y=98
29x23y=110

d. ax+by=ab
bxay=a+b

e. x+y=a+b
axby=a2b2

f. (ab)x+(a+b)y=a22abb2
(a+b)(x+y)=a2b2

g. 8x3y=5xy
5y=2xy

h. 3(2x+y)=7xy
3(x+3y)=11xy

i. 49x+51y=499
51x+49y=501

j. 217x+131y=913
131x+217y=827

Answer

1.
i. (3, 2)
ii. (3, -1)
iii. (8, 3)
iv. (1,-1)
v. (a, b)
vi. (a,-b)
vii. (0, 0) ,(22/31, 11/23)
viii. (0,0) (1,3/2)
ix. (1 ½ , 9/2)
x. 3,2


Qustion 2
Solve these linear equation in two variable ( x and y)
i. 12x+13y=2
13x+12y=136

ii) 2x+3y=9xy
4x+9y=21xy
Where x0,y0

iii. 22x+y+15xy=5
55x+y+45xy=14

iv.5x+y2xy=1
15x+y+7xy=10

v. bx+cy=a+b
ax(1ab1a+b)+cy(1ba1b+a)=2aa+b

vi)12(2x+3y)+17(3x2y)=1720
7(2x+3y)1(3x2y)=285

vii.x+12y+411=2
x+32+2y+317=5

viii.7x2yxy=5
8x+7yxy=15

ix. xa+yb=2
axby=a2b2

x.57x+y+6xy=5
38x+y+21xy=9

Answer

i. (1/2, 1/3)
Hint: Take 16x=p,16y=q and then solve in p& q and then find x and y

ii. (1 ,3)
Hint: Take 1x=p,1y=q and then solve in p& q and then find x and y

iii. (8,3)
Hint: Take 1x+y=p,1xy=q and then solve in p& q and then find x and y

iv. (3,2)
Hint: Take 1x+y=p,1xy=q and then solve in p& q and then find x and y

v. ab,bc vi. (2,1)
Hint: Take 12x+3y=p,13x2y=q and then solve in p& q and then find x and y

vii. 5, 7
viii. (1, 1)
Hint:
Convert into these forms
7y2x=5
8y+7y=15
Take 1x=p,1y=q and then solve in p& q and then find x and y

ix. (a,b)
x. (11, 8)



Question 3.
In a cyclic quadrilateral ABCD,Find the four angles.
a. A=(2x+4), B=(y+3), C=(2y+10), D=(4x5).
b. A=(2x1), B=(y+5), C=(2y+15) and D=(4x7)

Answer

a. A=(2x+4), B=(y+3), C=(2y+10), D=(4x5)
In a cyclic quadrilateral, Opposite angles are supplementary.
A+C=1800 and B+D=1800
So 2x+4+2y+10=180 or x+y=83
y+3+4x5=180 or y+4x=182
Solving the above equation by Substitution method
x=33 and y=50
So Angles are

700,530,1100,127 0
b) A=(2x1), B=(y+5), C=(2y+15) and D=(4x7)
Solving similarly, we get
650, 550, 1150, 1250


Question 4.
Given below are three equations. Two of them have infinite solutions and two have a unique solution. State the two pairs:
3x2y=4
6x+2y=4
9x6y=12

Answer

For Infinite solution:

We have a1a2=b1b2=c1c2 is the equation for infinite solution.
Equation 1 and 3 is satisfying the condition
3x2y=4; 9x6y=12
3/9 = -2/-6 = 4/12

This also means equation 1 and equation 3 are same

Unique solution:
We have a1/a2 is not equal to b1/b2 is the equation for the unique solution.
Equation 1 and 2 and Equation 2 and 3 is satisfying the condition
3x-2y=4; 6x+2y=4 & 6x+2y=4; 9x-6y=1
3/6 is not equal to -2/2 & 6/9 is not equal to 2/-6.


Question 5
Find the values of a and b for which the following system of linear equations has infinite number of solutions:
2x+3y=7
2ax+(a+b)y=28

Answer

We have
a1a2=b1b2=c1c2
22a=3a+b=728
Solving this , we get a=4 and b=8


Question 6
Find the values of a and b for which the following system of linear equations has infinite number of solutions:
2x3y=7
(a+b)x(a+b3)y=4a+b

Answer

We have
a1a2=b1b2=c1c2
2a+b=3a+b3=74a+b
2a+b=3a+b3
a+b+6=0 -(1)
3a+b3=74a+b
5a-4b+21=0 -(2)
Solving (1) and (2) by substitution method we get
a=-5,b=-1


Question 7.
In a ABC, A=x, B=(3x2), C=y Also CB=9. Find the three angles.

Answer

In a triangle,sum of angles is equal to 3600
So
x+(3x2)+y=180 or 4x+y=182 -(1)
Also given
CB=9 or y(3x2)=9 or y3x=7 or y=7+3x
Substituting this in (1)
4x+7+3x=182
7x=175
x=25
So angles are
25,73, 82


Question 8.
Find the value of k for which the system of equations x + 2y -3 = 0 and ky + 5x + 7 = 0 has a unique solution.

Answer

For unique solution
a1a2b1b2
Subsituting the values
152k
k10


Question 9
For what value of a the system of linear equations 2x+3y=7 and (a1)x+(a+1)y=3a1 represent parallel lines.

Answer

For parallel lines
a1a2=b1b2=c1c2
2a1=3a+1=73a1
or a=5


Question 10
If the lines x+2y+7=0 and 2x+ky+18=0 intersect at a point, then find the value of k.

Answer

For unique solution
a1a2b1b2
Subsituting the values
122k
k4


Question 11
Is x=5,y=5 a solution of the linear equation 3x+2y5=0?

Answer

3(5) +2(-5) -5=0
So x=5 and y=-5 is solution of the equation


Question 12.
The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. Find the present ages of the son and the father.

Answer

let Father age is y and son age is x
First condition
y=6x
Second condition
y+4=4(x+4)
So we have
6x+4=4x+16
x=6 and y=36



Summary

This linear equations in two variables worksheet class 10 with solutions is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.



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