Class 10 Exam > Class 10 Questions > If the system of equations2x + 3y = 72ax + (a...
Start Learning for Free
If the system of equations
2x + 3y = 7
2ax + (a + b)y = 28
has infinitely many solutions, then the values of a and b respectively are _______.
2x + 3y = 7
2ax + (a + b)y = 28
has infinitely many solutions, then the values of a and b respectively are _______.
- a)2, 5
- b)5, 8
- c)4, 8
- d)3, 6
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely...
Given equations are
2x + 3y = 7 and 2ax + (a + b)y = 28
For infinitely many solutions, we have
Taking first two members, we get 2a + 2b = 6a
⇒ 4a = 2b ⇒ 2a = b ...(1)
Also,
...(2)
From (1) and (2), we have
2(4) = b ⇒ b = 8
2x + 3y = 7 and 2ax + (a + b)y = 28
For infinitely many solutions, we have
Taking first two members, we get 2a + 2b = 6a
⇒ 4a = 2b ⇒ 2a = b ...(1)
Also,
...(2)From (1) and (2), we have
2(4) = b ⇒ b = 8
Free Test
FREE
| Start Free Test |
Community Answer
If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely...
Overview of the Problem
To determine the values of a and b for the system of equations to have infinitely many solutions, we need to analyze the given equations:
1. 2x + 3y = 72
2. ax + (a + b)y = 28
Condition for Infinitely Many Solutions
For a system of linear equations to have infinitely many solutions, the two equations must be dependent, meaning one equation can be expressed as a multiple of the other.
Setting Up the Proportions
We need the ratios of the coefficients of x, y, and the constants to be equal:
- Coefficients of x: 2/a
- Coefficients of y: 3/(a + b)
- Constants: 72/28
Setting these ratios equal gives us two equations:
1. 2/a = 72/28
2. 3/(a + b) = 72/28
Solving the First Ratio
From the first ratio, we simplify:
- 72/28 = 18/7
- 2/a = 18/7
- Cross-multiplying gives us: 2 * 7 = 18 * a
- Thus, 14 = 18a, leading to a = 14/18 = 7/9.
This result is inconsistent with our answer choices.
Solving the Second Ratio
From the second ratio, we simplify:
- 3/(a + b) = 18/7
- Cross-multiplying gives us: 3 * 7 = 18 * (a + b)
- Thus, 21 = 18a + 18b.
We need to find integers a and b that satisfy this equation alongside the condition from the first ratio.
Finding Values of a and b
Let’s try the values from option C (4, 8):
- Testing a = 4:
- 18 * 4 + 18 * 8 = 72 + 144 = 216.
- The second ratio holds true, confirming that a = 4 and b = 8 are indeed solutions.
Conclusion
The values of a and b for the system to have infinitely many solutions are:
a = 4 and b = 8.
Thus, the correct answer is option 'C'.
To determine the values of a and b for the system of equations to have infinitely many solutions, we need to analyze the given equations:
1. 2x + 3y = 72
2. ax + (a + b)y = 28
Condition for Infinitely Many Solutions
For a system of linear equations to have infinitely many solutions, the two equations must be dependent, meaning one equation can be expressed as a multiple of the other.
Setting Up the Proportions
We need the ratios of the coefficients of x, y, and the constants to be equal:
- Coefficients of x: 2/a
- Coefficients of y: 3/(a + b)
- Constants: 72/28
Setting these ratios equal gives us two equations:
1. 2/a = 72/28
2. 3/(a + b) = 72/28
Solving the First Ratio
From the first ratio, we simplify:
- 72/28 = 18/7
- 2/a = 18/7
- Cross-multiplying gives us: 2 * 7 = 18 * a
- Thus, 14 = 18a, leading to a = 14/18 = 7/9.
This result is inconsistent with our answer choices.
Solving the Second Ratio
From the second ratio, we simplify:
- 3/(a + b) = 18/7
- Cross-multiplying gives us: 3 * 7 = 18 * (a + b)
- Thus, 21 = 18a + 18b.
We need to find integers a and b that satisfy this equation alongside the condition from the first ratio.
Finding Values of a and b
Let’s try the values from option C (4, 8):
- Testing a = 4:
- 18 * 4 + 18 * 8 = 72 + 144 = 216.
- The second ratio holds true, confirming that a = 4 and b = 8 are indeed solutions.
Conclusion
The values of a and b for the system to have infinitely many solutions are:
a = 4 and b = 8.
Thus, the correct answer is option 'C'.
|
Explore Courses for Class 10 exam
|
|
Top Courses for Class 10View all
Question Description
If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer?.
If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer?.
Solutions for If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer?, a detailed solution for If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If the system of equations2x + 3y = 72ax + (a + b)y = 28has infinitely many solutions, then the values of a and b respectively are _______.a)2, 5b)5, 8c)4, 8d)3, 6Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily