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0
955
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Consider the system

−8x − 12y = 36
5y − 6x = 41

Find x.

Jun 3, 2014

#2
+24949
+7

(1) -8x - 12y = 36  ->   8x + 12y = -36

(2)  5y -   6x = 41  ->  -6x +   5y =  41

$$\boxed{\begin{array}{rcrcr} 8x & + & 12y &=& -36 \\ -6x & + & 5y &=& 41 \end{array}}$$

\boxed{\begin{array}{rcrcr}
8x & + & 12y &=& -36 \\
-6x & + & 5y &=&  41
\end{array}}

$$\\x= \frac{\begin{vmatrix} -36 & 12 \\ 41 & 5 \end{vmatrix}}{\begin{vmatrix} 8& 12 \\ -6 & 5 \end{vmatrix}} =\frac{-36*5-41*12}{8*5-(-6)*12} =\frac{-180-492}{40+72} =\frac{-672}{112}=-6\\ \boxed{x=-6} \\y= \frac{\begin{vmatrix} 8 & -36 \\ -6 & 41 \end{vmatrix}}{\begin{vmatrix} 8& 12 \\ -6 & 5 \end{vmatrix}} =\frac{8*41-(-6)(-36)}{8*5-(-6)*12} =\frac{328-216}{40+72} =\frac{112}{112}=1\\ \boxed{y=1}$$

\\x=
\frac{\begin{vmatrix}
-36 & 12 \\
41 & 5
\end{vmatrix}}{\begin{vmatrix}
8& 12 \\
-6 & 5
\end{vmatrix}}
=\frac{-36*5-41*12}{8*5-(-6)*12}
=\frac{-180-492}{40+72}
=\frac{-672}{112}=-6\\
\boxed{x=-6}
\\y=
\frac{\begin{vmatrix}
8 & -36 \\
-6 & 41
\end{vmatrix}}{\begin{vmatrix}
8& 12 \\
-6 & 5
\end{vmatrix}}
=\frac{8*41-(-6)(-36)}{8*5-(-6)*12}
=\frac{328-216}{40+72}
=\frac{112}{112}=1\\
\boxed{y=1}

Jun 4, 2014

#1
+111321
+5

−8x − 12y = 36
-6x +  5y  = 41         We can use the elimination method to solve this

We'll eliminate "x"   by multiplying the top equation by -6 on both sides and by multiplying the bottom equation by 8 on both sides   .....this gives us

48x + 72y = -216

-48x + 40y = 328       Now....add the equations together

112y = 112                So...it's clear that y = 1

To find "x,"  substitute 1 for y in any of the equations....I'll use -6x + 5y = 41

-6x + 5(1) = 41

-6x + 5 = 41        subtract 5 ftom both sides

-6x = 36             divide by -6 on both sides

x = -6

So....x = -6 and y = 1......you should verify that these "work" in the other equations...I think you will find that they do......

Jun 3, 2014
#2
+24949
+7

(1) -8x - 12y = 36  ->   8x + 12y = -36

(2)  5y -   6x = 41  ->  -6x +   5y =  41

$$\boxed{\begin{array}{rcrcr} 8x & + & 12y &=& -36 \\ -6x & + & 5y &=& 41 \end{array}}$$

\boxed{\begin{array}{rcrcr}
8x & + & 12y &=& -36 \\
-6x & + & 5y &=&  41
\end{array}}

$$\\x= \frac{\begin{vmatrix} -36 & 12 \\ 41 & 5 \end{vmatrix}}{\begin{vmatrix} 8& 12 \\ -6 & 5 \end{vmatrix}} =\frac{-36*5-41*12}{8*5-(-6)*12} =\frac{-180-492}{40+72} =\frac{-672}{112}=-6\\ \boxed{x=-6} \\y= \frac{\begin{vmatrix} 8 & -36 \\ -6 & 41 \end{vmatrix}}{\begin{vmatrix} 8& 12 \\ -6 & 5 \end{vmatrix}} =\frac{8*41-(-6)(-36)}{8*5-(-6)*12} =\frac{328-216}{40+72} =\frac{112}{112}=1\\ \boxed{y=1}$$

\\x=
\frac{\begin{vmatrix}
-36 & 12 \\
41 & 5
\end{vmatrix}}{\begin{vmatrix}
8& 12 \\
-6 & 5
\end{vmatrix}}
=\frac{-36*5-41*12}{8*5-(-6)*12}
=\frac{-180-492}{40+72}
=\frac{-672}{112}=-6\\
\boxed{x=-6}
\\y=
\frac{\begin{vmatrix}
8 & -36 \\
-6 & 41
\end{vmatrix}}{\begin{vmatrix}
8& 12 \\
-6 & 5
\end{vmatrix}}
=\frac{8*41-(-6)(-36)}{8*5-(-6)*12}
=\frac{328-216}{40+72}
=\frac{112}{112}=1\\
\boxed{y=1}

heureka Jun 4, 2014