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2a+b=12 and |b| less than or equal to 12

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2a+b=12 and |b| less than or equal to 12 [#permalink] New post   29 Feb 2024, 06:16
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\( 2a+b=12\) and \(|b| \leq 12\)

How many ordered pairs ab are solutions of the above system such that a and b both are integers?

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13


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E
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Re: 2a+b=12 and |b| less than or equal to 12 [#permalink] New post   30 Mar 2024, 11:19
1
If |b| is less than or equal to 12, then b is more than or equal to -12 or less than or equal to 12.
Also, b can only be an even number because 2a is always even.

Now, if I understand it correctly, the ordered pairs are

0,12 (0+12)
1,10 (2+10)
2,8 (4+8)
3,6 (6+6)
4,4 (8+4)
5,2 (10+2)
6,0 (12+0)
7, -2 (14-2)
8, -4 (16-4)
9, -6 (18-6)
10, -8 (20-8)
11, -10 (22-10)
12, -12 (24-12)

Is there a way to solve it algebraically?
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Re: 2a+b=12 and |b| less than or equal to 12 [#permalink] New post   30 Mar 2024, 11:26
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We see that the value of b ranges from −12 to +12; this follows that b can have 25 number of integer values.

Now let us see how many integer values a can have.

\(2a+b=12 \)

\(a=12-\frac{b}{2}\)

\(a=6-\frac{b}{2}\)

We see that for a to be an integer, b/2 must be an integer; this follows that b must be an even number.

Out of 25 possible values of b, 13 are even (including 0); so for a to be an integer, the set of arrangement can only have 13 ordered pairs.
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2a+b=12 and |b| less than or equal to 12 [#permalink] New post   17 Oct 2024, 17:33
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Given that 2a+b=12 and |b| ≤ 12 and we need to find how many ordered pairs ab are solutions of the above system such that a and b both are integers?

|b| ≤ 12
=> -12 ≤ b ≤ 12

2a + b = 12
=> 2a = 12 - b
=> a = \(\frac{12-b}{2}\)
=> Now 12 is Even and 2 is even and we know that Even / Even is an integer if Numerator is divisible by denominator
[ MASTER Even Odd Numbers link here ]

=> For a to be an integer b has to be an even number

=> We need to find the total number of even numbers between (and including) -12 and 12
=> -12, -10, -8,...., 8, 10, 12
= \(\frac{(12- (-12))}{2}\) + 1 [ MASTER Sequences link here ]
= 12 + 1 = 13

So, Answer will be E
Hope it helps!

MASTER Absolute Value Problems

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2a+b=12 and |b| less than or equal to 12 [#permalink]
17 Oct 2024, 17:33
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