Question 1203091
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Any linear equation in standard form is Ax+By = C
A,B,C are integers.


For example: 2x+3y = 4


If we have the 2x+3y portion, then plugging in (x,y) = (1,19) leads to:
2x+3y 
2*1+3*19 
2+57
59


This means 2x+3y = 59 when (x,y) = (1,19) 
Rephrased another way: (x,y) = (1,19) is a solution to 2x+3y = 59.
Another rephrasing: The point (1,19) is on the line 2x+3y = 59


Now let's generate the other equation.
Pick another two random integers.
Let's say we go for 5 and 6 to generate 5x+6y
Plug x = 1 and y = 19 into that expression
5x+6y
5*1+6*19
5+114
119


Therefore, 5x+6y = 119 when x = 1 and y = 19
In other words, the ordered pair (x,y) = (1,19) is a solution to 5x+6y = 119
The point (1,19) is on the line 5x+6y = 119.


We've demonstrated that (x,y) = (1,19) is a solution to both 2x+3y = 59 and 5x+6y = 119 simultaneously.
This point is on both lines at the same time.
This means the two lines intersect at (1,19).
Use Desmos, GeoGebra, TI83/84, or any graphing tool to visually confirm this.


A non-visual way to confirm is to use any of the following:<ul><li>Substitution</li><li>Elimination</li><li>Matrix Row Reduction (either REF or RREF)</li><li>Cramer's Rule</li></ul>As the tutor Theo has shown using Desmos, there is another system of equations that yields the solution (x,y) = (1,19)


It turns out there are infinitely many systems possible. I encourage you to explore another possible system.
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