Question 1143188
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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 [{{{A^-1}}}] then [x] = [{{{A^-1}}}][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:

 {{{ (matrix(2,2, 2,-1, 4,-1)) }}} x {{{ (matrix(2,1, L,M)) }}} = {{{ (matrix(2,1, 0, 30)) }}}

One would then find the inverse of the 2x2 matrix and that inverse, multiplied by the right hand side  {{{ (matrix(2,1, 0, 30)) }}} gives the answer.

The inverse of {{{ (matrix(2,2, 2,-1, 4,-1)) }}}  is {{{ (matrix(2,2, -1/2, 1/2, -2,1)) }}} 


{{{ (matrix(2,1,L,M))}}} = {{{ (matrix(2,2, -1/2, 1/2, -2,1)) }}} x {{{ (matrix(2,1, 0, 30)) }}} = {{{ (matrix(2,1, (-1/2)(0)+(1/2)(30), (-2)(0)+(1)(30))) }}} = {{{ (matrix(2,1, 15, 30)) }}}  --->  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).