Method of Elimination

Q.

A fraction becomes $\frac{4}{5}$ if one is added to both numerator and denominator. If, however, 5 is subtracted from numerator and denominator, the resultant fraction becomes $\frac{1}{2}$. Find the fraction.

1. $\frac{-7}{9}$

2. $\frac{7}{10}$

3. $\frac{-7}{10}$

4. $\frac{7}{9}$

Q. A two - digit number is 7 times the sum of its digits.The number formed by reversing the digits is 18 less than the original number.Find the number.

1. 25
2. 24
3. 43
4. 42

Q. The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

1. 5, 3
2. 7, 4
3. 9, 6
4. 10, 7

Q.

The sum of a two digit number and the number obtained by reversing the digits is 121 and the two digits differ by 3. Find the number.

1. 62 or 26

2. 47 or 74

3. 41 or 14

4. 87 or 87

Q.

The sum of digits of a two-digit number is $13.$ If the number is subtracted from the one obtained by reversing the digits, the result is $45.$ Find the number.

1. $94$
2. $49$
3. $85$
4. $76$

Q. Question 13
Write an equation of a line passing through the point representing solution of the pair of linear equations x + y = 2 and 2x – y = 1, how many such lines can we find?

Q. Solve the following system of linear equations by using the method of elimination by equating the coefficients
$\sqrt{3}x-\sqrt{2}y=\sqrt{3};\sqrt{5}x+\sqrt{3}y=\sqrt{2}$

Q.

The sum of two numbers is 209. If one number is 7 less than two times of the other, then find the two numbers.

1. 137 and 72

2. 133 and 74

3. 147 and 62

4. 127 and 82

Q.

Find the solution of the given system of equations.

$\frac{x+y-8}{2}=\frac{x+2y-14}{8}=\frac{3x+y-12}{11}$

1. (1, -1)

2. (2, 6)

3. (2, 2)

4. (0, 1)

Q. The standard form of a linear equation in two variables x and y, whose respective coefficients are 2 and -3 and has the constant as 4 is ____.

1. $3x-2y+4=0$
2. $2x-3y-4=0$
3. $3x-2y-4=0$
4. $2x-3y+4=0$

Q.

Question 39

State whether the following statement is true (T) or false (F).

Two different equations can never have the same answer.

Q.

Solve the two equations using method of elimination.

$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$

1. x = $\frac{(b_1{c}_2-b_2{c}_1)}{(a_1{b}_2-a_2{b}_1)}$, y = $\frac{(c_1{a}_2-c_2{a}_1)}{(a_1{b}_2-a_2{b}_1)}$

2. x = $\frac{(b_2{c}_1-b_1{c}_2)}{(a_1{b}_2-a_2{b}_1)}$, y = $\frac{(c_1{a}_1-c_2{a}_2)}{(a_1{b}_2-a_2{b}_1)}$

3. x = $\frac{(b_2{c}_2-b_1{c}_1)}{(a_1{b}_2-a_2{b}_1)}$, y = $\frac{(c_2{a}_2-c_1{a}_1)}{(a_1{b}_2-a_2{b}_1)}$

4. x = $\frac{(b_1{c}_2+b_2{c}_1)}{(a_1{b}_2+a_2{b}_1)}$, y = $\frac{(c_1{a}_2+c_2{a}_1)}{(a_1{b}_2+a_2{b}_1)}$

Q. Solve:

Q.

From Bangalore bus stand, if we buy 2 tickets to Malleshwaram and 3 tickets to Yeshwanthpur, the total cost is ₹ 46; but if we buy 3 tickets to Malleshwaram and 5 tickets to Yeshwanthpur, then the total cost is ₹ 74. Find the fares from Bangalore bus stand to Malleshwaram and to Yeshwanthpur respectively.

1. ₹12 and ₹ 10

2. ₹ 10 and ₹ 8

3. ₹ 10 and ₹ 12

4. ₹ 8 and ₹ 10

Q.

The sum of the digits of a two-digit number is 9. If 27 is added to it, the digits are reversed. Find the number.

1. 27

2. 63

3. 36

4. 72

Q. We have two equations
$3x–4y=1$ ---------- (1)
$5x–6y=7$ ---------- (2)
In the above equations, what operation should be carried out to make the coefficients of $x$ equal?

1. Multiply equation (1) by 6 and equation (2) by 4
2. Multiply equation (1) by 3 and equation (2) by 5
3. Multiply equation (1) by 5 and equation (2) by 3
4. Multiply equation (1) by 8 and equation (2) by 3

Q.

Solve

13x + 11y = 70

11x + 13y = 74

1. x = 2 and y = 4

2. x = -2 and y = -4

3. x = 2 and y = 6

4. x = 3 and y = 4

Q. Use the Factor Theorem to determine whether $g\left(x\right)$ is a factor of $p\left(x\right)$ in each of the following cases:
(i) $p\left(x\right)=2x^3+x^2-2x-1,g\left(x\right)=x+1$
(ii) $p\left(x\right)=x^3+3x^2+3x+1,g\left(x\right)=x+2$
(iii) $p\left(x\right)=x^3-4x^2+x+6,g\left(x\right)=x-3$

Q.

The sum of a two digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number.

1. 132

2. 39

3. 48

4. 88

Q. The sum of a numerator and denominator of a fraction is $18$. If the denominator increased by $2$, the fraction reduces to $\frac{1}{3}$. Find the fraction.

Q.

Find the solution of the given pair of equations
$2x+3y=9,3x+4y=5$

1. x = 21, y = 7
2. x = - 21, y = 17
3. x = 19, y = 14
4. x = 20, y = 12

Q.

x + 2y = 5 and 4x + 4y - 6 = 0 are equations of two lines which intersect at $(-3,\frac{5}{3})$

1. True

2. False

Q.

$3x–4y=1$...(1)
$5x–6y=7$...(2)
In the above equations, which of the following step can to be done to numerically equate the coefficients of $x$?

1. Multiply equation (1) by 3 and equation (2) by 5.

2. Multiply equation (1) by 6 and equation (2) by 4.

3. Multiply equation (1) by 5 and equation (2) by 3.

4. Multiply equation (1) by 4 and equation (2) by 6.

Q. The difference between the ages of two sisters is$10$ years. If $15$ years ago, the elder one was twice $10$ years. If $15$ years ago, the elder one was twice as old as the younger one, find their present ages.

Q. Solve equations using substitution method:
$5x-2y=10$ and $4x-6y=3$

1. $\frac{-27}{11}$ and $\frac{25}{22}$
2. $\frac{27}{11}$ and $\frac{-25}{22}$
3. $\frac{27}{11}$ and $\frac{25}{22}$
4. $\frac{-27}{11}$ and $\frac{-25}{22}$

Q. Solve the following pair of equations by the elimination method and the substitution method:

1. $x=\frac{19}{5},y=\frac{6}{5}$
2. $x=\frac{24}{7},y=\frac{6}{5}$
3. $x=\frac{13}{2},y=\frac{6}{5}$
4. $x=\frac{17}{7},y=\frac{6}{5}$

Q.

If twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178 and thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177, then find the smallest even number.

1. 4

2. 6

3. 8

4. 2

Q.

If twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178 and thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177, then the difference between the sum of 5 consecutive odd numbers and 4 consecutive even numbers is

1. 1

2. 3

3. 7

4. 5

Q.

If twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178 and thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177, then find the middle number when all the numbers are arranged in ascending order.

1. 9

2. 8

3. 7

4. 6

Q.

If twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178. and thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177, then if all the 9 numbers are arranged in ascending order , the sum of the smallest and largest number is

1. 15

2. 16

3. 18

4. 17