xy+4x=2880
xy-12y=2160
x and y in both equations is the same
\(\begin{array}{|lrcll|} \hline (1) & xy+4x &=& 2880 \\ (2) & xy-12y &=& 2160 \\ \hline \\ (1)-(2): & xy+4x -(xy-12y)&=& 2880-2160 \\ & xy+4x -xy+12y &=& 720 \\ & 4x+12y &=& 720 \quad & | \quad : 4 \\ & x+3y &=& 180 \\ (3) & x &=& 180-3y \\ \hline \end{array}\)
(3) in (2):
\(\begin{array}{|rcll|} \hline (2) & xy-12y &=& 2160 \\ & y\cdot (x-12) &=& 2160 \quad & | \quad x = 180-3y \\ & y\cdot (180-3y-12) &=& 2160 \\ & 180y-3y^2-12y &=& 2160 \\ & -3y^2+168y &=& 2160 \quad & | \quad : (-3) \\ & y^2-56y &=&-720 \\ & y^2-56y +720 &=& 0 \\ & (y-36)(y-20) &=& 0 \\\\ & y_1 = 36 &\text{or}& \quad y_2 = 20 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline (3) & x &=& 180-3y \\ & x_1 &=& 180-3\cdot 36 \\ & x_1 &=& 72 \\\\ & x_2 &=& 180-3\cdot 20 \\ & x_2 &=& 120 \\ \hline \end{array}\)
Solutions: (72,36) and (120,20)
xy + 4x= 2880 → x (y + 4) = 2880 → x = 2880/ (y + 4) (1)
xy - 12y= 2160 (2)
Sub (1) into (2)
[2880/ (y + 4)] y - 12y = 2160 multiply through by ( y + 4)
2880y - 12y( y + 4) = 2160( y + 4) simplify
2880y - 12y^2 - 48y = 2160y + 8640
12y^2 - 672 y + 8640 = 0
y^2 - 56y + 720 = 0 factor
( y - 36) (y - 20) = 0
Set each factor to 0 and y = 36 or y = 20
When y = 36, x = 2880/[36 + 4] = 2880/40 = 72
When y = 20, x = 2880/[20 + 4] = 2880/24 = 120
Solutions (72, 36) and (120, 20 )
xy+4x=2880
xy-12y=2160
x and y in both equations is the same
\(\begin{array}{|lrcll|} \hline (1) & xy+4x &=& 2880 \\ (2) & xy-12y &=& 2160 \\ \hline \\ (1)-(2): & xy+4x -(xy-12y)&=& 2880-2160 \\ & xy+4x -xy+12y &=& 720 \\ & 4x+12y &=& 720 \quad & | \quad : 4 \\ & x+3y &=& 180 \\ (3) & x &=& 180-3y \\ \hline \end{array}\)
(3) in (2):
\(\begin{array}{|rcll|} \hline (2) & xy-12y &=& 2160 \\ & y\cdot (x-12) &=& 2160 \quad & | \quad x = 180-3y \\ & y\cdot (180-3y-12) &=& 2160 \\ & 180y-3y^2-12y &=& 2160 \\ & -3y^2+168y &=& 2160 \quad & | \quad : (-3) \\ & y^2-56y &=&-720 \\ & y^2-56y +720 &=& 0 \\ & (y-36)(y-20) &=& 0 \\\\ & y_1 = 36 &\text{or}& \quad y_2 = 20 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline (3) & x &=& 180-3y \\ & x_1 &=& 180-3\cdot 36 \\ & x_1 &=& 72 \\\\ & x_2 &=& 180-3\cdot 20 \\ & x_2 &=& 120 \\ \hline \end{array}\)
Solutions: (72,36) and (120,20)