Answer:
(x,y)=(5,8)
y=8
x=5
Step-by-step explanation:
Method 1 ; Substitution Method
[tex]\begin{bmatrix}y=2x-2\\ y=3x-7\end{bmatrix}\\\\\mathrm{Subsititute\:}y=3x-7\\\\\begin{bmatrix}3x-7=2x-2\end{bmatrix}\\\\Isolate\:x \:for\:3x -7=2x-2\:\:,x=5\\\\\mathrm{For\:}y=3x-7\\\\\mathrm{Subsititute\:}x=5\\\\y=3\times\:5-7\\\\Simplify\\\\y=8\\\\y=8,\:x=5[/tex]
Method 2 : Elimination Method
[tex]\begin{bmatrix}y=2x-2\\ y=3x-7\end{bmatrix}\\\\\mathrm{Arrange\:equation\:variables\:for\:elimination}\\\\\begin{bmatrix}y-2x=-2\\ y-3x=-7\end{bmatrix}\\\\y-3x=-7\\-\\\underline{y-2x=-2}\\-x=-5\\\\\begin{bmatrix}y-2x=-2\\ -x=-5\end{bmatrix}\\\\Solve\:-x=-5\:for\:x \:,x =5\\\\\mathrm{For\:}y-2x=-2\mathrm{\:plug\:in\:}x=5\\\\Solve \:y-2\times 5 =-2\:for\:y\:\: ,y =8\\\\y=8,\:x=5[/tex]
Method 3 : Gaussian's Elimination
[tex]\begin{bmatrix}y=2x-2\\ y=3x-7\end{bmatrix}\\\\\mathrm{Arrange\:equation\:variables\:for\:elimination}\\\\\begin{bmatrix}y-2x=-2\\ y-3x=-7\end{bmatrix}\\\\\mathrm{Write\:a\:matrix\:with\:the\:coefficients\:and\:solutions}\\\\\begin{bmatrix}1&-2&¦&-2\\ 1&-3&¦&-7\end{bmatrix}\\\\Reduce \:matrix\:to\:echelon\:form\\\\\begin{bmatrix}1&0&¦&8\\ 0&1&¦&5\end{bmatrix}\\\\y=8,\:x=5[/tex]
