document.write( "Question 1143188: 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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Algebra.Com's Answer #763968 by math_helper(2461)\"\" \"About 
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document.write( "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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document.write( "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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document.write( "Let M=Mike's age, and  L=Liza's age\r
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document.write( "M = 2L         (1)   (from \"Now he is only twice as old as Liza\")
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document.write( "M-10 = 4(L-10) (2)   (from \"Ten years ago, Mike was four times as old as Liza.\")\r
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document.write( "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%5E-1\"] then [x] = [\"A%5E-1\"][b]\r
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document.write( "Substitution:
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document.write( "(1) says M=2L so we can write (2) as:
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document.write( "2L-10 = 4(L-10)
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document.write( "2L-10 = 4L-40   distributed the 4 into parenthesis
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document.write( "   30 = 2L      subtracted 2L from each side, added 40 to each side
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document.write( "   15 = L       divided each side by 2\r
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document.write( "L=15 tells us M=30  (by plugging L=15 into (1))\r
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document.write( "Subtraction of equations:
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document.write( "(1) - (2) gives:\r
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document.write( "M-(M-10) = 2L-4(L-10)
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document.write( "M-M+10 = 2L-4L+40
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document.write( "    10 = -2L+40
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document.write( "   -30 = -2L
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document.write( "    15 = L    (same result as substitution, as expected)\r
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document.write( "The matrix method comes from Linear Algebra:\r
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document.write( "(1) can be written     2L - M = 0
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document.write( "(2) can be written     4L - M = 30\r
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document.write( "Taking the coefficients we can form the matrix equation:\r
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document.write( " \"+%28matrix%282%2C2%2C+2%2C-1%2C+4%2C-1%29%29+\" x \"+%28matrix%282%2C1%2C+L%2CM%29%29+\" = \"+%28matrix%282%2C1%2C+0%2C+30%29%29+\"\r
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document.write( "One would then find the inverse of the 2x2 matrix and that inverse, multiplied by the right hand side  \"+%28matrix%282%2C1%2C+0%2C+30%29%29+\" gives the answer.\r
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document.write( "The inverse of \"+%28matrix%282%2C2%2C+2%2C-1%2C+4%2C-1%29%29+\"  is \"+%28matrix%282%2C2%2C+-1%2F2%2C+1%2F2%2C+-2%2C1%29%29+\" \r
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document.write( "\"+%28matrix%282%2C1%2CL%2CM%29%29\" = \"+%28matrix%282%2C2%2C+-1%2F2%2C+1%2F2%2C+-2%2C1%29%29+\" x \"+%28matrix%282%2C1%2C+0%2C+30%29%29+\" =  = \"+%28matrix%282%2C1%2C+15%2C+30%29%29+\"  --->  L=15, M=30\r
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document.write( "Check:\r
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document.write( "L=15, M=30, obviously now Mike is twice as old as Liza.  But let's check their ages 10 years ago:\r
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document.write( "Liza was 15-10 = 5
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document.write( "Mike was 30-10 = 20   (20 is 4*5 so this checks out).\r
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