For infinitely many solutions of 5x + (r + 1)y = 11 and 15x + 18y = 33, what is r?
Answer and explanation
Correct answer: r = 5
For infinitely many solutions, the two equations must represent the same line. Therefore, the ratios of corresponding coefficients and constants must be equal. Here, 5/15 = 1/3 and 11/33 = 1/3. The remaining ratio must satisfy (r + 1)/18 = 1/3. Multiplying both sides by 18 gives r + 1 = 6, so r = 5. Substituting r = 5 makes the first equation 5x + 6y = 11, while multiplying it by 3 gives 15x + 18y = 33, exactly the second equation. Thus the equations are coincident and have infinitely many common solutions. Option B is correct. Every other listed value breaks the proportionality condition and cannot produce infinitely many solutions.
Frequently asked questions
What is the correct answer to this question?
r = 5
Why is this the correct answer?
For infinitely many solutions, the two equations must represent the same line. Therefore, the ratios of corresponding coefficients and constants must be equal. Here, 5/15 = 1/3 and 11/33 = 1/3. The remaining ratio must satisfy (r + 1)/18 = 1/3. Multiplying both sides by 18 gives r + 1 = 6, so r = 5. Substituting r = 5 makes the first equation 5x + 6y = 11, while multiplying it by 3 gives 15x + 18y = 33, exactly the second equation. Thus the equations are coincident and have infinitely many common solutions. Option B is correct. Every other listed value breaks the proportionality condition and cannot produce infinitely many solutions.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.
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