```Question 94795

x+y=11 Use Cramer's rule to solve the x-y=5 following system of equations. If D=0, use another method to solve the system.
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To solve the system of equations by Cramer's rule:

{{{Ax+By=C}}}
{{{Px+Qy=R}}}

D is the matrix of coefficients as they appear
on the left of the equal sign:

D = {{{(matrix(2,2,A,B,P,Q))}}} = {{{AQ-BP}}}

D<sub>x</sub> is the matrix which is like D except that,
since x is the FIRST unknown letter, it has
its FIRST column {{{matrix(2,1,A,P)}}} replaced
by the column of numbers on the right side of
the equal signs in the original system, {{{matrix(2,1,C,R)}}}.
So

D<sub>x</sub> = {{{(matrix(2,2,C,B,R,Q))}}} = {{{CQ-BR}}}

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D<sub>y</sub> is the matrix which is like D except that,
since y is the SECOND unknown letter, it has
its SECOND column {{{matrix(2,1,B,Q)}}} replaced
by the column of numbers on the right side of
the equal signs in the original system, {{{matrix(2,1,C,R)}}}.
So

D<sub>y</sub> = {{{(matrix(2,2,A,C,P,R))}}} = {{{AR-CP}}}

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Then x = {{{Dx/D}}} and y = {{{Dy/D}}}

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So for your system:

x + y = 11
x - y =  5

Write that as

{{{1x+1y=11}}}
{{{1x-1y=5}}}

D is the matrix of coefficients as they appear
on the left of the equal sign:

D = {{{(matrix(2,2,1,1,1,-1))}}} = {{{(1)(-1)-(1)(1)}}} = {{{-1-1}}} = {{{-2}}}

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D<sub>x</sub> is the matrix which is like D except that,
since x is the FIRST unknown letter, it has
its FIRST column {{{matrix(2,1,1,1)}}} replaced
by the column of numbers on the right side of
the equal signs in the original system, {{{matrix(2,1,11,5)}}}.
So

D<sub>x</sub> = {{{(matrix(2,2,11,1,5,-1))}}} = {{{(11)(-1)-(1)(5)}}} = {{{-11-5}}} = {{{-16}}}

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D<sub>y</sub> is the matrix which is like D except that,
since y is the SECOND unknown letter, it has
its SECOND column {{{matrix(2,1,1,-1)}}} replaced
by the column of numbers on the right side of
the equal signs in the original system, {{{matrix(2,1,11,5)}}}.
So

D<sub>y</sub> = {{{(matrix(2,2,1,11,1,5))}}} = {{{(1)(5)-(11)(1)}}} = {{{5-11}}} = {{{-6}}}

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Then x = {{{Dx/D}}} = {{{(-16)/(-2)}}} = {{{8}}} and y = {{{Dy/D}}} = {{{(-6)/(-2)}}} = {{{3}}}.

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Edwin</pre>```