On comparing the ratios a 1 a 2 , b 1 b 2 and c 1 c 2 , find out whether 2 x - 2 y = 2 and 4 x - 4 y = 5 are consistent or inconsistent.

Given: A pair of linear equations are 2 x - 2 y = 2 and 4 x - 4 y = 5 . Writing the equations of the lines in the standard form, 2 x - 2 y - 2 = 0 ........ 1 4 x - 4 y - 5 = 0 ......... 2 Comparing the equations ( 1 ) and ( 2 ) with a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 , we can write, a 1 = 2 , b 1 = - 2 , c 1 = - 2 a 2 = 4 , b 2 = - 4 , c 2 = - 5 Finding the ratios of the coefficients, a 1 a 2 = 2 4 = 1 2 , b 1 b 2 = - 2 - 4 = 1 2 and c 1 c 2 = - 2 - 5 = 2 5 We can see that, a 1 a 2 = b 1 b 2 ≠ c 1 c 2 . The given pair of linear equations have no solution. Hence, the given pair of linear equations is inconsistent.

There are some students in the two examination halls A and B . To make the number of students equal in each hall, 10 students are sent from A to B but, if 20 students are sent from B to A , the number of students in A becomes double the number of students in B , then find the number of students in the both halls.

Let the number of students in halls A and B are x and y , respectively. Now, by given condition, x - 10 = y + 10 ⇒ x − y = 20 . . . ( i ) and x + 20 = 2 y - 20 . . . ( i i ) On subtracting Eq. ( i i ) from Eq. ( i ) , we get ( x − y ) − ( x − 2 y ) = 20 + 60 x - y - x + 2 y = 80 ⇒ y = 80 On Putting y = 80 in Eq. i , we get x - 80 = 20 ⇒ x = 100 and y = 80 Hence, 100 students are in hall A and 80 students are in hall B

The pair of equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 are consistent, then

a 1 a 2 ≠ b 1 b 2 and a 1 a 2 = b 1 b 2 = c 1 c 2

MEDIUM

10th Andhra Pradesh Board

IMPORTANT

Earn 100

The larger of two supplementary angles exceeds the smaller by $18 °$ . Find the angles.

Important Questions on Pair of Linear Equations in Two Variables

HARD

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 18

On comparing the ratios

\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}

and

\frac{c_{1}}{c_{2}}

, find out whether

2 x - 2 y = 2 and 4 x - 4 y = 5

are consistent or inconsistent.

MEDIUM

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 19

There are some students in the two examination halls

A

and

B

. To make the number of students equal in each hall,

10

students are sent from

A

B

but, if

20

students are sent from

B

A

, the number of students in

A

becomes double the number of students in

B

, then find the number of students in the both halls.

MEDIUM

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 20

The difference between two numbers is

26

and one number is three times the other. Find the numbers.

HARD

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 21

Yash scored $40$ marks in a test, getting $3$ marks for each right answer and losing $1$ marks for each wrong answer. If $4$ marks had been awarded for each correct answer and $2$ marks been deducted for each incorrect answer, then Yash would have scored $50$ marks. How many questions were there in the test?

MEDIUM

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 22

Two numbers are in the ratio $5 : 6$ . If $8$ is subtracted from each of the numbers, the ratio becomes $4 : 5$ , then find the numbers.

EASY

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 23

By comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}$ , find out whether the lines represented by the following pair of linear equations intersect at a point, are parallel or are coincident.

$5 x - 4 y + 8 = 0$

$7 x + 6 y - 9 = 0$

EASY

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 24

$9 x + 3 y + 12 = 0$

$18 x + 6 y + 24 = 0$

MEDIUM

10th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Pair of Linear Equations in Two Variables>Exercise>Q 25

'When $5$ is subtracted from length and breadth of the rectangle, the perimeter becomes $26$ .' What is the mathematical form of the statement?

The larger of two supplementary angles exceeds the smaller by 18°. Find the angles.

Important Questions on Pair of Linear Equations in Two Variables

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The larger of two supplementary angles exceeds the smaller by $18 °$ . Find the angles.