ten yrs. ago, mike was four times as old as liza. Now he is only twice as old as Liza. How old are they now?

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Ten years ago, Mike was four times as old as Liza.  Now he is only twice as old as Liza.  How old are they now?

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Let M=Mike's age, and  L=Liza's age



M = 2L         (1)   (from "Now he is only twice as old as Liza")

M-10 = 4(L-10) (2)   (from "Ten years ago, Mike was four times as old as Liza.")



There are several approaches to solve this system of equations.  You couild subsititue 2L for M (specified by equation(1)) into (2).  You could also subtract the entire equation (2) from equation (1).  A third option is to set up a matrix and find its inverse, then use that inverse to solve: for [A][x]=[b] find [] then [x] = [][b]







Substitution:

(1) says M=2L so we can write (2) as:

2L-10 = 4(L-10)

2L-10 = 4L-40   distributed the 4 into parenthesis

   30 = 2L      subtracted 2L from each side, added 40 to each side

   15 = L       divided each side by 2



L=15 tells us M=30  (by plugging L=15 into (1))





Subtraction of equations:

(1) - (2) gives:



M-(M-10) = 2L-4(L-10)

M-M+10 = 2L-4L+40

    10 = -2L+40

   -30 = -2L

    15 = L    (same result as substitution, as expected)







The matrix method comes from Linear Algebra:



(1) can be written     2L - M = 0

(2) can be written     4L - M = 30



Taking the coefficients we can form the matrix equation:



  x  = 



One would then find the inverse of the 2x2 matrix and that inverse, multiplied by the right hand side   gives the answer.



The inverse of   is  





 =  x  =  =   --->  L=15, M=30





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Check:



L=15, M=30, obviously now Mike is twice as old as Liza.  But let's check their ages 10 years ago:



Liza was 15-10 = 5

Mike was 30-10 = 20   (20 is 4*5 so this checks out).



 

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