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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
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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19 Mar 2019, 01:29
00:00

Difficulty:

65% (hard)

Question Stats:

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

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Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. Then a^2 -b^2 + c^2 is

(A) 0
(B) 1
(C) 4
(D) 7
(E) 8

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e-GMAT Representative
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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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.

_________________
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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