The second equation is (2) times the first so both lines are coincident. Coincident lines have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. The second equation is (2) times the first so both lines are coincident. Coincident lines have infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है इसलिए दोनों रेखाएं संपाती हैं। संपाती रेखाओं में अनंत हल होते हैं।
Here (3/6=4/8) but (11/25) is different so there is no solution. In exams equal first two ratios and a different third ratio mean inconsistent.
Step 2
Why this answer is correct
The correct answer is C. असंगत / Inconsistent. Here (3/6=4/8) but (11/25) is different so there is no solution. In exams equal first two ratios and a different third ratio mean inconsistent.
Step 3
Exam Tip
यहां (3/6=4/8) लेकिन (11/25) अलग है इसलिए कोई हल नहीं। परीक्षा में पहले दो अनुपात बराबर और तीसरा अलग हो तो असंगत लिखें।
Here \(1/2 \ne 7/9\) so the lines intersect at one point. Different coefficient ratios give one unique solution.
Step 2
Why this answer is correct
The correct answer is D. एक अद्वितीय हल है / There is one unique solution. Here \(1/2 \ne 7/9\) so the lines intersect at one point. Different coefficient ratios give one unique solution.
Step 3
Exam Tip
यहां \(1/2 \ne 7/9\) इसलिए रेखाएं एक बिंदु पर कटती हैं। अलग गुणांक अनुपात एक अद्वितीय हल देते हैं।
When all three ratios are equal both equations give the same line. This is called a consistent and dependent pair.
Step 2
Why this answer is correct
The correct answer is A. संगत और आश्रित / Consistent and dependent. When all three ratios are equal both equations give the same line. This is called a consistent and dependent pair.
Step 3
Exam Tip
तीनों अनुपात बराबर होने पर दोनों समीकरण एक ही रेखा देते हैं। इसे संगत और आश्रित युग्म कहते हैं।
Equal first two ratios give the same slope and a different third ratio keeps the lines separate. Such lines have no solution.
Step 2
Why this answer is correct
The correct answer is B. समानांतर और अलग / Parallel and distinct. Equal first two ratios give the same slope and a different third ratio keeps the lines separate. Such lines have no solution.
Step 3
Exam Tip
पहले दो अनुपात बराबर होने से ढाल समान होती है और तीसरा अलग होने से रेखाएं अलग रहती हैं। ऐसी रेखाओं का कोई हल नहीं होता।
When the ratios of (a) and (b) are different the lines intersect. Intersecting lines give exactly one common point.
Step 2
Why this answer is correct
The correct answer is C. एक अद्वितीय हल / One unique solution. When the ratios of (a) and (b) are different the lines intersect. Intersecting lines give exactly one common point.
Step 3
Exam Tip
जब (a) और (b) के अनुपात अलग होते हैं तो रेखाएं कटती हैं। कटती रेखाएं एक ही सामान्य बिंदु देती हैं।
The second equation is (2) times the first so both lines are the same. All points on the same line are solutions.
Step 2
Why this answer is correct
The correct answer is B. अनंत हल / Infinitely many solutions. The second equation is (2) times the first so both lines are the same. All points on the same line are solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है इसलिए दोनों रेखाएं एक ही हैं। एक ही रेखा के सभी बिंदु हल होते हैं।
Here (4/8=1/2) but (6/15) is different so the lines are parallel. Distinct parallel lines give no solution.
Step 2
Why this answer is correct
The correct answer is A. कोई हल नहीं / No solution. Here (4/8=1/2) but (6/15) is different so the lines are parallel. Distinct parallel lines give no solution.
Step 3
Exam Tip
यहां (4/8=1/2) लेकिन (6/15) अलग है इसलिए रेखाएं समानांतर हैं। समानांतर अलग रेखाएं हल नहीं देतीं।
C. अनंत हल हैं/There are infinitely many solutions
Step 1
Concept
The first equation is (3) times the second so both are the same line. In such questions spotting the common multiplier is the fastest method.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल हैं / There are infinitely many solutions. The first equation is (3) times the second so both are the same line. In such questions spotting the common multiplier is the fastest method.
Step 3
Exam Tip
पहला समीकरण दूसरे का (3) गुना है इसलिए दोनों एक ही रेखा हैं। ऐसे प्रश्नों में समान गुणक पहचानना तेज तरीका है।
Here (2/4=(-3)/(-6)) but (5/7) is different. Therefore the lines are parallel and distinct.
Step 2
Why this answer is correct
The correct answer is C. समानांतर अलग रेखाएं / Distinct parallel lines. Here (2/4=(-3)/(-6)) but (5/7) is different. Therefore the lines are parallel and distinct.
Step 3
Exam Tip
यहां (2/4=(-3)/(-6)) लेकिन (5/7) अलग है। इसलिए रेखाएं समानांतर और अलग हैं।
Here \(7/5 \ne 2/4\) so the lines meet at one point. One intersection means one unique solution.
Step 2
Why this answer is correct
The correct answer is D. एक अद्वितीय हल / One unique solution. Here \(7/5 \ne 2/4\) so the lines meet at one point. One intersection means one unique solution.
Step 3
Exam Tip
यहां \(7/5 \ne 2/4\) इसलिए रेखाएं एक बिंदु पर मिलती हैं। एक intersection का अर्थ एक अद्वितीय हल है।
Completely overlapping lines are coincident. Every point on them satisfies both equations.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. Completely overlapping lines are coincident. Every point on them satisfies both equations.
Step 3
Exam Tip
पूरी तरह मिलने वाली रेखाएं संपाती होती हैं। उनके प्रत्येक बिंदु से दोनों समीकरण संतुष्ट होते हैं।
One common point gives one unique solution. Therefore it is a consistent and independent pair.
Step 2
Why this answer is correct
The correct answer is A. संगत और स्वतंत्र / Consistent and independent. One common point gives one unique solution. Therefore it is a consistent and independent pair.
Step 3
Exam Tip
एक सामान्य बिंदु होने से एक अद्वितीय हल मिलता है। इसलिए यह संगत और स्वतंत्र युग्म है।
The second equation is (3) times the first so both lines are coincident. Coincident lines give infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is D. अनंत हल / Infinitely many solutions. The second equation is (3) times the first so both lines are coincident. Coincident lines give infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (3) गुना है इसलिए दोनों रेखाएं संपाती हैं। संपाती रेखाएं अनंत हल देती हैं।
Here (9/3=6/2) but (21/8) is different so the lines are parallel. Distinct parallel lines have no solution.
Step 2
Why this answer is correct
The correct answer is B. कोई हल नहीं / No solution. Here (9/3=6/2) but (21/8) is different so the lines are parallel. Distinct parallel lines have no solution.
Step 3
Exam Tip
यहां (9/3=6/2) लेकिन (21/8) अलग है इसलिए रेखाएं समानांतर हैं। समानांतर अलग रेखाओं में कोई हल नहीं होता।
Here \(3/6 \ne 5/11\) so the lines intersect. Intersecting lines have exactly one common solution.
Step 2
Why this answer is correct
The correct answer is A. एक अद्वितीय हल / One unique solution. Here \(3/6 \ne 5/11\) so the lines intersect. Intersecting lines have exactly one common solution.
Step 3
Exam Tip
यहां \(3/6 \ne 5/11\) इसलिए रेखाएं कटती हैं। कटती रेखाओं का केवल एक सामान्य हल होता है।
Having no common solution identifies an inconsistent pair. In a graph it appears as distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is C. असंगत / Inconsistent. Having no common solution identifies an inconsistent pair. In a graph it appears as distinct parallel lines.
Step 3
Exam Tip
कोई सामान्य हल न होना असंगत युग्म की पहचान है। ग्राफ में यह अलग समानांतर रेखाओं जैसा दिखता है।
Infinitely many common solutions occur only when both equations represent the same line. In exams this can be written as coincident lines.
Step 2
Why this answer is correct
The correct answer is B. एक ही रेखा / Same line. Infinitely many common solutions occur only when both equations represent the same line. In exams this can be written as coincident lines.
Step 3
Exam Tip
अनंत सामान्य हल तभी मिलते हैं जब दोनों समीकरण एक ही रेखा दर्शाते हैं। परीक्षा में इसे coincident lines लिख सकते हैं।
Here (2/6=1/3) but (4/10) is different so the lines are parallel. Such lines never meet.
Step 2
Why this answer is correct
The correct answer is C. समानांतर अलग रेखाएं / Distinct parallel lines. Here (2/6=1/3) but (4/10) is different so the lines are parallel. Such lines never meet.
Step 3
Exam Tip
यहां (2/6=1/3) लेकिन (4/10) अलग है इसलिए रेखाएं समानांतर हैं। ऐसी रेखाएं कभी नहीं मिलतीं।
C. एक बिंदु पर कटती रेखाएं/Lines intersecting at one point
Step 1
Concept
Here \(4/8 \ne 7/13\) so the slopes are different. Lines with different slopes intersect at one point.
Step 2
Why this answer is correct
The correct answer is C. एक बिंदु पर कटती रेखाएं / Lines intersecting at one point. Here \(4/8 \ne 7/13\) so the slopes are different. Lines with different slopes intersect at one point.
Step 3
Exam Tip
यहां \(4/8 \ne 7/13\) इसलिए ढालें अलग हैं। अलग ढाल वाली रेखाएं एक बिंदु पर कटती हैं।
In the first option the second equation is (2) times the first. Therefore the lines are coincident and have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is A. (x+y=5) और (2x+2y=10) / (x+y=5) and (2x+2y=10). In the first option the second equation is (2) times the first. Therefore the lines are coincident and have infinitely many solutions.
Step 3
Exam Tip
पहले विकल्प में दूसरा समीकरण पहले का (2) गुना है। इसलिए दोनों रेखाएं संपाती हैं और अनंत हल हैं।
In the second option (1/2=2/4) but (7/15) is different. Therefore the lines are parallel and have no solution.
Step 2
Why this answer is correct
The correct answer is B. (x+2y=7) और (2x+4y=15) / (x+2y=7) and (2x+4y=15). In the second option (1/2=2/4) but (7/15) is different. Therefore the lines are parallel and have no solution.
Step 3
Exam Tip
दूसरे विकल्प में (1/2=2/4) लेकिन (7/15) अलग है। इसलिए रेखाएं समानांतर हैं और कोई हल नहीं है।
In the third option \(4/2 \ne 1/3\) so the lines intersect. Intersecting lines give one unique solution.
Step 2
Why this answer is correct
The correct answer is C. (4x+y=8) और (2x+3y=7) / (4x+y=8) and (2x+3y=7). In the third option \(4/2 \ne 1/3\) so the lines intersect. Intersecting lines give one unique solution.
Step 3
Exam Tip
तीसरे विकल्प में \(4/2 \ne 1/3\) इसलिए रेखाएं कटती हैं। कटती रेखाएं एक अद्वितीय हल देती हैं।
An inconsistent pair has no common solution. Identify it in a graph by distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is B. उसमें कोई हल नहीं होता / It has no solution. An inconsistent pair has no common solution. Identify it in a graph by distinct parallel lines.
Step 3
Exam Tip
असंगत युग्म में कोई सामान्य हल नहीं होता। इसे ग्राफ में अलग समानांतर रेखाओं से पहचानें।
A. उसमें कम से कम एक हल होता है/It has at least one solution
Step 1
Concept
A consistent pair may have one or infinitely many solutions. The key point is that at least one common solution must exist.
Step 2
Why this answer is correct
The correct answer is A. उसमें कम से कम एक हल होता है / It has at least one solution. A consistent pair may have one or infinitely many solutions. The key point is that at least one common solution must exist.
Step 3
Exam Tip
संगत युग्म में एक या अनंत हल हो सकते हैं। मुख्य बात है कि कम से कम एक सामान्य हल होना चाहिए।
The lines of a consistent and independent pair intersect at one point. Therefore there is only one unique solution.
Step 2
Why this answer is correct
The correct answer is C. एक अद्वितीय हल / One unique solution. The lines of a consistent and independent pair intersect at one point. Therefore there is only one unique solution.
Step 3
Exam Tip
संगत और स्वतंत्र युग्म की रेखाएं एक बिंदु पर कटती हैं। इसलिए केवल एक अद्वितीय हल होता है।
In a consistent and dependent pair both equations represent the same line. Therefore every common point is a solution.
Step 2
Why this answer is correct
The correct answer is D. अनंत हल / Infinitely many solutions. In a consistent and dependent pair both equations represent the same line. Therefore every common point is a solution.
Step 3
Exam Tip
संगत और आश्रित युग्म में दोनों समीकरण एक ही रेखा दर्शाते हैं। इसलिए हर सामान्य बिंदु हल होता है।
A. पहला दूसरे का (4) गुना है/The first is (4) times the second
Step 1
Concept
Multiplying (2x-y=3) by (4) gives (8x-4y=12). Therefore there are infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is A. पहला दूसरे का (4) गुना है / The first is (4) times the second. Multiplying (2x-y=3) by (4) gives (8x-4y=12). Therefore there are infinitely many solutions.
Step 3
Exam Tip
(2x-y=3) को (4) से गुणा करने पर (8x-4y=12) मिलता है। इसलिए अनंत हल हैं।
The second equation is (2) times the first. Therefore both equations represent the same line.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. The second equation is (2) times the first. Therefore both equations represent the same line.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है। इसलिए दोनों समीकरण समान रेखा दर्शाते हैं।
C. रेखाएं एक बिंदु पर कटती हैं/Lines intersect at one point
Step 1
Concept
Here \(13/4 \ne 2/5\) so the lines are not parallel. They will intersect at one point and give one solution.
Step 2
Why this answer is correct
The correct answer is C. रेखाएं एक बिंदु पर कटती हैं / Lines intersect at one point. Here \(13/4 \ne 2/5\) so the lines are not parallel. They will intersect at one point and give one solution.
Step 3
Exam Tip
यहां \(13/4 \ne 2/5\) इसलिए रेखाएं समानांतर नहीं हैं। वे एक बिंदु पर कटेंगी और एक हल देंगी।
Same slope and same intercept make both lines identical. Therefore infinitely many solutions are obtained.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. Same slope and same intercept make both lines identical. Therefore infinitely many solutions are obtained.
Step 3
Exam Tip
समान ढाल और समान अवरोध से दोनों रेखाएं एक ही हो जाती हैं। इसलिए अनंत हल मिलते हैं।
Same slope and different intercepts make the lines parallel and distinct. Therefore there is no common solution.
Step 2
Why this answer is correct
The correct answer is B. असंगत / Inconsistent. Same slope and different intercepts make the lines parallel and distinct. Therefore there is no common solution.
Step 3
Exam Tip
समान ढाल और अलग अवरोध से रेखाएं समानांतर और अलग होती हैं। इसलिए कोई सामान्य हल नहीं होता।
Lines with different slopes meet at exactly one point. Therefore the pair is consistent and independent.
Step 2
Why this answer is correct
The correct answer is C. एक अद्वितीय हल / One unique solution. Lines with different slopes meet at exactly one point. Therefore the pair is consistent and independent.
Step 3
Exam Tip
अलग ढाल वाली रेखाएं एक ही बिंदु पर मिलती हैं। इसलिए युग्म संगत और स्वतंत्र होता है।
C. अनंत हल हैं/There are infinitely many solutions
Step 1
Concept
The first equation is (3) times the second. Therefore both lines are coincident and have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल हैं / There are infinitely many solutions. The first equation is (3) times the second. Therefore both lines are coincident and have infinitely many solutions.
Step 3
Exam Tip
पहला समीकरण दूसरे का (3) गुना है। इसलिए दोनों रेखाएं संपाती हैं और अनंत हल हैं।
