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▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
▪️For which value(s) for λ , do the pair of the given linear equations
λx + y = λ as well as x + λy = 1 have
(i) no solution?
(ii) infinitely many solutions?
▪️For which value(s) for λ , do the pair of the given linear equations
λx + y = λ as well as x + λy = 1 have
(i) no solution?
(ii) infinitely many solutions?
Most Upvoted Answer
▪️Half the perimeter of a rectangular garden, whose length is 4 m more...
Rectangular Garden Dimensions:
- Let the width of the garden be x meters.
- The length of the garden is then (x + 4) meters.
- The perimeter of a rectangle is given by P = 2(length + width).
Given, half the perimeter is 36 m:
- 2((x + 4) + x) = 36
- 2(2x + 4) = 36
- 4x + 8 = 36
- 4x = 28
- x = 7
Therefore, the width of the garden is 7 meters and the length is 7 + 4 = 11 meters.
Linear Equations:
(i) No Solution:
- For no solution, the two equations must represent parallel lines.
- To find parallel lines, the slopes of the two equations should be equal.
- The slope of the first equation is -λ and the slope of the second equation is -1/λ.
- For no solution, -λ = -1/λ should not have a real solution.
- This means λ should not be equal to ±1.
(ii) Infinitely Many Solutions:
- For infinitely many solutions, the two equations must represent the same line.
- This happens when the two equations are multiples of each other.
- Multiply the first equation by -1/λ to make the coefficients of x in both equations equal.
- -1/λ(λx + y) = -1/λ(λ)
- -x/λ - y = -1
- Compare this with the second equation x + λy = 1.
- The two equations are multiples of each other when 1/λ = λ.
- This gives two possible values for λ: λ = 1 and λ = -1.
- Let the width of the garden be x meters.
- The length of the garden is then (x + 4) meters.
- The perimeter of a rectangle is given by P = 2(length + width).
Given, half the perimeter is 36 m:
- 2((x + 4) + x) = 36
- 2(2x + 4) = 36
- 4x + 8 = 36
- 4x = 28
- x = 7
Therefore, the width of the garden is 7 meters and the length is 7 + 4 = 11 meters.
Linear Equations:
(i) No Solution:
- For no solution, the two equations must represent parallel lines.
- To find parallel lines, the slopes of the two equations should be equal.
- The slope of the first equation is -λ and the slope of the second equation is -1/λ.
- For no solution, -λ = -1/λ should not have a real solution.
- This means λ should not be equal to ±1.
(ii) Infinitely Many Solutions:
- For infinitely many solutions, the two equations must represent the same line.
- This happens when the two equations are multiples of each other.
- Multiply the first equation by -1/λ to make the coefficients of x in both equations equal.
- -1/λ(λx + y) = -1/λ(λ)
- -x/λ - y = -1
- Compare this with the second equation x + λy = 1.
- The two equations are multiples of each other when 1/λ = λ.
- This gives two possible values for λ: λ = 1 and λ = -1.
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▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions?
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▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about ▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions?.
▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about ▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ▪️Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.▪️For which value(s) for λ , do the pair of the given linear equationsλx + y = λ as well as x + λy = 1 have(i) no solution?(ii) infinitely many solutions?.
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