+130551 A little more on this strange graph.....
Apparently, if this can be factored into this form....(x + y) (x - y) = 0, we have a graph of intersecting lines...let's see...
2x^2 - 3y^2 = 5xy
2x^2 - 5xy - 3y^2 = 0
(2x + y) (x - 3y) = 0
Since this "reducible" to this form, this is a degenerate conic that will form two intersecting lines.
Here's the graph, again.....https://www.desmos.com/calculator/hwmmfcew1u
Notice something......if we set the first term in the above factorization to 0, we have 2x + y = 0, or just y = -2x...and this is the line ine on the graph that "falls" from right to left!!! Similarly, doing the same thing to the second factored term produces y = (1/3)x.....and this is the other line on the graph that "rises" from left to right....!!!
And notice one last thing......just like we might do in a quadratic by "factoring and setting to 0" to find the roots....we are doing something similar here....except that, instead of generating "roots," we're generating equations of lines....!!!!
Here's some more info about these odd graphs....http://www.digplanet.com/wiki/Degenerate_conic
+130551 2x^2-3y^2=5xy and-3x+y=5 rearranging the second equation, we have y = 3x + 5
And putting this into the first equation, we have
2x^2 - 3(3x + 5)^2 = 5x(3x + 5) simplify
2x^2 - 3(9x^2 + 30x +25) = 15x^2 + 25x
2x^2 -27x^2 - 90x - 75 = 15x^2 + 25x
-13x^2 - 27x^2 - 115x - 75 = 0 multiply through by -1
13x^2 + 27x^2 + 115x + 75 = 0
40x^2 + 115x + 75 = 0 divide through by 5
8x^2 + 23x + 15 = 0 factor
(x + 1 )(8x + 15) = 0
And setting each factor to 0, we have that x = -1 and x = -15/8
And using y = 3x + 5 when x = -1, y = 2 and when x = -15/8, y = -5/8
So....our solutions are (-1, 2) and (-15/8, -5/8)
Here's a graph.......https://www.desmos.com/calculator/mdhdt8ytqq
The "blue" function is a straight line.....the other one is something I haven't seen before....a pair of intersecting lines that are "rotated"....very odd !!!!
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P.S. - I found an internet source that says the strange graph is known as a "degenerate" conic....!!!!
+130551 x-y=3 and xy-5x+y=-13
Using the first equation we can rearrange it as y = x -3 and substituting this in the second, we have
x(x -3) - 5x + (x -3) = -13 simplify
x ^2 - 3x - 5x + x -3 = -13
x^2 - 7x + 10 = 0 factor
(x -5) ( x-2) = 0 and setting each factor to 0, we have that x = 5 and x =2
And using y = x -3, when x = 5, y = 2 and when x = 2 , y = -1
So...our solutions (intersection points) are (5,2) and (2, -1)
Here's the graph of both equations.....https://www.desmos.com/calculator/3gd27vhg4y
The first graph is a line and the second is a "rotated" conic (in this case, a hyperbola)
+130551 x^2+y^2=24 and y=2x+3
Substituting for y in the first equation, we have
x^2 + (2x + 3)^2 = 24 simplify
x^2 + 4x^2 + 12x + 9 = 24
5x^2 + 12x - 15 = 0 this won't factor..so using the onsite solver, we have
$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{15}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{111}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}\right)}{{\mathtt{5}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{111}}}}{\mathtt{\,-\,}}{\mathtt{6}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{3.307\: \!130\: \!750\: \!570\: \!547\: \!8}}\\
{\mathtt{x}} = {\mathtt{0.907\: \!130\: \!750\: \!570\: \!547\: \!8}}\\
\end{array} \right\}$$
Let's round these to -3.31 and .91
And using y = 2x+ 3, when x = -3.31, y = -3.614 and when x =.91, y = 4.814
So the solutions are (-3.31, -3.614) and ( .91, 4.814)
This is just the intersection of a line and a circle...here's the graph....https://www.desmos.com/calculator/oz2dnf8ems
