Bill and Frank 12/12/2009 Here's what's right: 1. you returned to the problem after probably finding the original tootorial a bit much 2. you decided to try and understand the material by creating a Bill and Frank problem which was similar to the Tooni and Zee problem Here's what's wrong: 1. your 3rd equation should read f1 = 3b2 2. your 5th equation has a typo. it should read f0 - f2 = 6 3. your 6th equation is wrong. it should read f1 - f0 = b1 - b0 4. your chart showing the columns of ages does not correspond to your English description of the age relationships. In the chart, the future age of Frank (which turns out is actually a past age - see below) is supposed to be 3 times a past age of Bill; your red arrow connecting a 6-year-old Frank and a 6-year-old Bill does not have this relationship. 5. in your 2-equation 2-unknown example, you make an error in the substitution step 2(17-f) + 6f = 54 is correct i'm not sure how you got the next line. because it is in error, you end up concluding that f=2, which is wrong (you decided that f would be 5 so this is what you should get). the next line should read 34 - 2f + 6f = 54 Next lines should be 34 + 4f = 54 4f = 20 f = 5 6. there's a typo in your 2nd equation in the 2 equation 2 unknown example. it should read 2b + 6f = 54 7. the matrix you claim is the inverse, namely [6 -1] [-2 1] is not quite right. i'm not sure where you got this matrix from (did you use the online inverter?) you correctly express your problem in matrix form, namely [1 1][b] = [17] [2 6][f] = [54] if you multiply [6 -1][1 1] [-2 1][2 6] you get [4 0] [0 4] which is equal to 4 times [1 0] [0 1] i didn't mention this but what a regular number times a matrix means is you multiply every element in the matrix by the number. so the inverse matrix is 1/4 times your matrix. the solution to your problem is 1/4 [6 -1] [17] = [b] [-2 1] [54] [f] which works out to 1/4 ( (6)(17) + (-1)(54) ) = 1/4 (102 - 54) = 1/4 (48) = 12 = b and 1/4 ( (-2)(17) + (1)(54) ) = 1/4 (-34 + 54) = 1/4 (20) = 5 = f Ok. Let's do the full problem. Note Well! Because this problem not only requires that the sum of the current ages is 17 but also has two other requirements involving Bill / Frank age relationships, we can't expect Bill and Frank will end up being 12 and 5 respectively any more. The equations are b0 + f0 = 17 equation 1 b0 = 2f1 equation 2 f1 = 3b2 equation 3 (note my correction) b0 - b2 = 6 equation 4 f0 - f2 = 6 equation 5 (note my correction) f1 - f0 = b1 - b0 equation 6 (note my correction) We use legal algebraic manipulations to turn this into 1b0 + 0b1 + 0b2 + 1f0 + 0f1 + 0f2 = 17 1b0 + 0b1 + 0b2 + 0f0 - 2f1 + 0f2 = 0 0b0 + 0b1 - 3b2 + 0f0 + 1f1 + 0f2 = 0 1b0 + 0b1 - 1b2 + 0f0 + 0f1 + 0f2 = 6 0b0 + 0b1 + 0b2 + 1f0 + 0f1 - 1f2 = 6 1b0 - 1b1 + 0b2 - 1f0 + 1f1 + 0f2 = 0 Using the online matrix inverter, we find the inverse to be 0.0 -0.2 -0.4 1.2 0.0 0.0 -1.0 -1.0 -1.0 3.0 0.0 -1.0 0.0 -0.2 -0.4 0.2 0.0 0.0 1.0 0.2 0.4 -1.2 0.0 0.0 0.0 -0.6 -0.2 0.6 0.0 0.0 1.0 0.2 0.4 -1.2 -1.0 0.0 applying this to the RHS, the solution to your problem is [ 0.0 -0.2 -0.4 1.2 0.0 0.0][17] [b0] [-1.0 -1.0 -1.0 3.0 0.0 -1.0][ 0] [b1] [ 0.0 -0.2 -0.4 0.2 0.0 0.0][ 0] = [b2] [ 1.0 0.2 0.4 -1.2 0.0 0.0][ 6] [f0] [ 0.0 -0.6 -0.2 0.6 0.0 0.0][ 6] [f1] [ 1.0 0.2 0.4 -1.2 -1.0 0.0][ 0] [f2] The most important ones to expand out are rows 1 and 4 0(17) -.2(0) -.4(0) +1.2(6) +0(6) +0(0) = 7.2 = b0 1(17) +.2(0) +.4(0) -1.2(6) +0(6) +0(0) = 9.8 = f0 The chart looks like b0 7.2 9.8 f0 6.2 5.2 4.2 3.2 2.2 b2 1.2 3.8 f2 b1 1.0 3.6 f1 So ... it looks like the tenses in your English are not correct! Bill's and Frank's ages add up to 17 and Bill is twice as old as Frank was when Frank was three times as old as Bill was six years ago. How old is Bill?