(a+b) (2+i) = b+1 + (10+2a)i
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Step-by-step explanation:
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Solve the following complex equation: '(a + b)(2 + i) = b + 1 + (10 + 2a)i' to find values for 'a' and 'b'
The first step with this question is to expand all the brackets; this will produce the following equation:
2a + ai + 2b + bi = b + 1 + 10i + 2ai
The next step is to put all the terms which contain what you're trying to work out (in this case 'a' and 'b') on the left side of the equation and the rest on the right. At this point you can ignore whether there is an 'i' in the term, just focus on the 'a's and 'b's. This will produce:
2a + ai + 2b + bi - b - 2ai = 1 + 10i
Now we focus on the complex element of this question and organise each side of the equation into the form X + Yi, in other words grouping all the imaginary terms (containing i's) and non-imaginary terms (not containing i's) separately. This gives:
(2a + b) + (ai + bi - 2ai) = 1 + 10i
The second bracket on the left side of the equation can be simplified as there is both an 'ai' term and a '-2ai' term which can be combined. The imaginary bracket can also be simplified by factorising out the 'i'. After carrying out this simplification you have the following equation:
(2a + b) + (b - a)i = 1 + 10i
You can now create a set of simultaneous equations by equating the imaginary terms on each side to each other and the same with the non-imaginary terms. These can then be solved to find 'a' and 'b', the simultaneous equations in this case are:
'2a + b = 1' (from the non-imaginary terms) and 'b - a = 10' (from the imaginary terms, you do not need to include the 'i's as they are on both sides and therefore would cancel instantly)
The second simultaneous equation can be rearranged to make 'b' the subject as follows: 'b = 10 + a' which can then be used as a substitution in the first simultaneous equation removing b and leaving only 1 unknown:
2a + (10 + a) = 1
3a + 10 = 1
3a = -9
a = -3
Substitute this value back into either of the simultaneous equations to produce a value for b:
b = 10 + a
b = 10 - 3
b = 7
If you're ever unsure of your answers, substitute them back into the original equation and check the two sides match:
(7-3)(2+i) = 7 + 1 + (10 + 2(-3))i
4(2 + i) = 8 + 4i
8 + 4i = 8 + 4i
Which confirms a = -3 and b = 7 is the correct answer