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What is the maximum value of P = 2x + 5y subject to the constraints x

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Bunuel
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What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   23 Apr 2018, 23:55
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What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14
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C
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Tulkin987
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   24 Apr 2018, 00:16
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Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


\(3x - y \geq 1\) --> \(y \leq 3x-1\)
\(3x + y \leq 5\) --> \(y \leq 5-3x\)

From above inequalities we know that \(3x-1=5-3x\) --> \(x=1\) and \(y \leq 5-3*1=2\)

For \(x=1\) and \(y=2\) maximum value of \(P = 2x + 5y = 2+10 = 12\)

Answer: C
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   24 Apr 2018, 02:45
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Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


The favorable region formed by the inequalities is a triangle with vertexes as A \((\frac{1}{2},\frac{1}{2})\); B \((\frac{5}{4}, \frac{5}{4})\) and C (1,2)

Now for P to be max, we can replace x,y as point C

Hence P = 2*1 + 2*5 = 12

Hence C
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   24 Apr 2018, 02:56
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Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


Answer is E,
we solved that at above Y<= 2, so 5y<=10. x<=y, so 2x<=2y it means that 2x<=10. Therefore, 2x+5y <= 10+4 = 14
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   30 Apr 2018, 08:11
huytm1912

y=2 and x=2 does not satisfy the stem 3x+y≤5
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   24 May 2018, 16:56
Tulkin987 wrote:
Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


\(3x - y \geq 1\) --> \(y \leq 3x-1\)
\(3x + y \leq 5\) --> \(y \leq 5-3x\)

From above inequalities we know that \(3x-1=5-3x\) --> \(x=1\) and \(y \leq 5-3*1=2\)

For \(x=1\) and \(y=2\) maximum value of \(P = 2x + 5y = 2+10 = 12\)

Answer: C



Hi Tulkin987
Why you converted the inequality to equality in the last 2 lines?
I mean we still need to say that y≤2 and x≥1
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   24 May 2018, 21:37
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hisho wrote:
Tulkin987 wrote:
Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


\(3x - y \geq 1\) --> \(y \leq 3x-1\)
\(3x + y \leq 5\) --> \(y \leq 5-3x\)

From above inequalities we know that \(3x-1=5-3x\) --> \(x=1\) and \(y \leq 5-3*1=2\)

For \(x=1\) and \(y=2\) maximum value of \(P = 2x + 5y = 2+10 = 12\)

Answer: C



Hi Tulkin987
Why you converted the inequality to equality in the last 2 lines?
I mean we still need to say that y≤2 and x≥1


\(3x - y \geq 1\) --> \(y \leq 3x-1\) and
\(3x + y \leq 5\) --> \(y \leq 5-3x\)
together imply that \(3x-1=5-3x\) and \(y \leq 2\)

Thus, \(x=1\) and \(y=2\).

Hope this helps.
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What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   16 Sep 2019, 01:24
Bunuel wrote:
What is the maximum value of \(P = 2x + 5y\) subject to the constraints \(x \leq y\), \(3x - y \geq 1\), and \(3x + y \leq 5\) ?

(A) 3.5
(B) 8.75
(C) 12
(D) 12.25
(E) 14


Condition:

\(x \leq y\)

we see inequalities are given in opposite direction thus we can deduct negative one from positive one.

\(3x - y -3x - y \leq 1 - 5\)

\(-2y \leq -4\)

\(y \geq 2\)

y ( mx) = 2

what about x?

we are looking for max value of the given equation thus we will try to maximize both x and y.

y=2

but x must not be 2 although it is given that \(x \leq y\).


\(3x + y \leq 5\)

This condition compels us to keep the value of x = 1.


\(P = 2x + 5y\)

P = 2*1 + 5*2 = 12.

C is the correct answer.
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink] New post   29 Mar 2024, 18:49
Can Bunuel post the official solution?
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Re: What is the maximum value of P = 2x + 5y subject to the constraints x [#permalink]
29 Mar 2024, 18:49
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