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Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b

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Math Expert
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Joined: 02 Sep 2009
Posts: 59725
Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b  [#permalink]

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New post 19 Mar 2019, 01:29
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

40% (03:14) correct 60% (02:55) wrong based on 20 sessions

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e-GMAT Representative
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Joined: 04 Jan 2015
Posts: 3158
Re: Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b  [#permalink]

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New post 19 Mar 2019, 01:39
1

Solution


Given:
    • The numbers a, b and c are real numbers
    • a – 7b + 8c = 4
    • 8a + 4b – c = 7

To find:
    • The value of the expression \(a^2 – b^2 + c^2\)

Approach and Working:
We know that
    • a – 7b + 8c = 4
    Or, a + 8c = 7b + 4 … (1)

    • Similarly, 8a + 4b – c = 7
    Or, 8a – c = 7 – 4b … (2)

Squaring both sides of equation (1) and (2) and then adding,
    • \(a^2 + 64c^2 + 16ac + 64a^2 + c^2 – 16ac = 49b^2 + 16 + 56b + 49 + 16b^2 – 56b\)
    Or, \(65a^2 + 65c^2 = 65b^2 + 65\)
    Or, \(a^2 – b^2 + c^2 = 1\)

Hence, the correct answer is option B.

Answer: B

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Re: Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b   [#permalink] 19 Mar 2019, 01:39
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Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b

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