For some real numbers a and b, the system of equations x+y=4 and (a+5)x+(b²-15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is:
Question:
For some real numbers a and b, the system of equations x+y=4 and (a+5)x+(b²-15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is:
For some real numbers a and b, the system of equations x+y=4 and (a+5)x+(b²-15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is:
Options
Answer: 33
Explanation:
For infinitely many solutions the two equations must be proportional. So there exists λ with (a + 5) = λ, (b² − 15) = λ, 8b = 4λ. From 8b = 4λ we get λ = 2b. Hence a + 5 = 2b and b² − 15 = 2b. Solve b² − 2b − 15 = 0 → (b − 5)(b + 3) = 0 → b = 5 or b = −3. If b = 5 → a = 2·5 − 5 = 5 → ab = 25. If b = −3 → a = 2(−3) − 5 = −11 → ab = (−11)(−3) = 33. Maximum possible value of ab = 33.
Explanation:
For infinitely many solutions the two equations must be proportional. So there exists λ with (a + 5) = λ, (b² − 15) = λ, 8b = 4λ. From 8b = 4λ we get λ = 2b. Hence a + 5 = 2b and b² − 15 = 2b. Solve b² − 2b − 15 = 0 → (b − 5)(b + 3) = 0 → b = 5 or b = −3. If b = 5 → a = 2·5 − 5 = 5 → ab = 25. If b = −3 → a = 2(−3) − 5 = −11 → ab = (−11)(−3) = 33. Maximum possible value of ab = 33.
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