Answer:
(x,y)=(5,3)
Step-by-step explanation:
Substitution Method
[tex]\begin{bmatrix}4x+y=23\\ 3x-y=12\end{bmatrix}\\\\Isolate \:and\: solve\:for\:x :\\4x+y=23 =x=\frac{23-y}{4}\\\\\mathrm{Substitute\:}x=\frac{23-y}{4}\\\\\begin{bmatrix}3\cdot \frac{23-y}{4}-y=12\end{bmatrix}\\\\Simplify\\\\\begin{bmatrix}\frac{69-7y}{4}=12\end{bmatrix}\\\\Isolate \:and\: solve\:for\:y :\\\\\frac{69-7y}{4}=12\\\\y =3\\\\\mathrm{For\:}x=\frac{23-y}{4}\\\\\mathrm{Substitute\:}y=3\\\\x=\frac{23-3}{4}\\\\\frac{23-3}{4} =5\\\\x=5,\:y=3[/tex]
Elimination Method
[tex]\begin{bmatrix}4x+y=23\\ 3x-y=12\end{bmatrix}\\\\\mathrm{Multiply\:}4x+y=23\mathrm{\:by\:}3\:\\\mathrm{:}\:\quad \:12x+3y=69\\\\\mathrm{Multiply\:}3x-y=12\mathrm{\:by\:}4\:\\\mathrm{:}\:\quad \:12x-4y=48\\\\\begin{bmatrix}12x+3y=69\\ 12x-4y=48\end{bmatrix}\\\\12x-4y=48\\-\\\underline{12x+3y=69}\\-7y=-21\\\\Solve :-7y=-21\\\\\mathrm{Divide\:both\:sides\:by\:}-7\\\frac{-7y}{-7}=\frac{-21}{-7}\\\\Simplify\\y=3\\\\\mathrm{For\:}12x+3y=69,\\\mathrm{\:plug\:in\:}y=3\\[/tex]
[tex]Solve :12x+3\cdot \:3=69\\\\12x+9=69\\\\12x+9-9=69-9\\\\Simplify\\\\12x=60\\\\\mathrm{Divide\:both\:sides\:by\:}12\\\\\frac{12x}{12}=\frac{60}{12}\\\\Simplify\\\\x=5\\\\\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}\\\\x=5,\:y=3[/tex]
