Question 235993
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Let *[tex \Large x] represent the smaller number.  Let *[tex \Large y] represent the larger.


Three times the smaller (*[tex \Large 3x]) increased by (*[tex \Large +]) 5 times the larger (*[tex \Large 5y]) is (=) 229.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ +\ 5y\ =\ 229]


Call that Equation 1, then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x\ -\ 25\ =\ 3y]


Put the second equation into standard form, that is *[tex \LARGE Ax\ +\ By\ =\ C]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x\ -\ 3y\ = 25]


And call that equation 2.  Multiply equation 1 by 3 and equation 2 by 5:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 9x\ +\ 15y\ =\ 687]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 20x\ -\ 15y\ = 125]


Notice that the multiplication factors were chosen so that the coefficients on the *[tex \Large y] terms are additive inverses. Add the like terms in the two equations, eliminating the *[tex \Large y] term:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 29x\ +\ 0y\ = 812]


Solve for *[tex \Large x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 28]


Substitute back into either original equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3(28)\ +\ 5y\ =\ 229]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5y\ =\ 145]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 29]


Check:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4(28)\ -\ 25\ =^?\ 3(29)]


You do the check arithmetic.



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