Question 273731
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Let *[tex \Large x] represent the age of the woman.  Let *[tex \Large y] represent the age of the daughter.  Then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ y\ +\ 24]


Which can be written:


<b>1:</b>  *[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ y\ =\ 24]


The woman's age two years ago: *[tex \Large x\ -\ 2].  The daughter's age two years ago:  *[tex \Large y\ -\ 2]


So we know:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 2\ =\ 4(y\ -\ 2)]


which can be written


<b>2:</b>  *[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 4y\ =\ -6]


Now that you have two linear equations in two variables, solve the system.  Multiply either equation by -1 and use the elimination method to derive a single-variable equation in *[tex \Large y]. Solve for *[tex \Large y] and then use that value to calculate *[tex \Large x].


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Let *[tex \Large x] represent the amount James has at the start.  Let *[tex \Large y] represent the amount John has at the start.  Then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 15y]


Which can be written:


<b>1:</b>  *[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 15y\ =\ 0]


After James gives John 12 pence John has *[tex \Large x\ -\ 12] remaining, and James now has *[tex \Large y\ +\ 12].  Furthermore we know:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 12\ =\ 3(y\ +\ 12)]


Which can be written:


<b>2:</b>  *[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 3y\ =\ 48]


Again, multiplying either equation by -1 sets up elimination of the *[tex \Large x] variable when the equations are added.  Solve for *[tex \Large y] and then use that value to calculate *[tex \Large x].



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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