SOLUTION: Hi need help please in solving all values of x and y and can you please show me each step this is another one 1. x^2 - 3y^2 = 13 x - 2y = 1

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Question 188362: Hi need help please in solving all values of x and y and can you please show me each step this is another one
1. x^2 - 3y^2 = 13
x - 2y = 1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x+-+2y+=+1 Start with the second equation.


x=+1%2B2y Add 2y to both sides to isolate "x".


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x%5E2+-+3y%5E2+=+13 Move onto the first equation


%281%2B2y%29%5E2+-+3y%5E2+=+13 Plug in x=+1%2B2y


1%2B4y%2B4y%5E2-+3y%5E2+=+13 FOIL


1%2B4y%2B4y%5E2-+3y%5E2+-+13=0 Subtract 13 from both sides.


y%5E2%2B4y+-+12=0 Combine like terms.



Notice we have a quadratic equation in the form of ay%5E2%2Bby%2Bc where a=1, b=4, and c=-12


Let's use the quadratic formula to solve for y


y+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


y+=+%28-%284%29+%2B-+sqrt%28+%284%29%5E2-4%281%29%28-12%29+%29%29%2F%282%281%29%29 Plug in a=1, b=4, and c=-12


y+=+%28-4+%2B-+sqrt%28+16-4%281%29%28-12%29+%29%29%2F%282%281%29%29 Square 4 to get 16.


y+=+%28-4+%2B-+sqrt%28+16--48+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-12%29 to get -48


y+=+%28-4+%2B-+sqrt%28+16%2B48+%29%29%2F%282%281%29%29 Rewrite sqrt%2816--48%29 as sqrt%2816%2B48%29


y+=+%28-4+%2B-+sqrt%28+64+%29%29%2F%282%281%29%29 Add 16 to 48 to get 64


y+=+%28-4+%2B-+sqrt%28+64+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


y+=+%28-4+%2B-+8%29%2F%282%29 Take the square root of 64 to get 8.


y+=+%28-4+%2B+8%29%2F%282%29 or y+=+%28-4+-+8%29%2F%282%29 Break up the expression.


y+=+%284%29%2F%282%29 or y+=++%28-12%29%2F%282%29 Combine like terms.


y+=+2 or y+=+-6 Simplify.


So the answers for "y" are y+=+2 or y+=+-6


Now simply plug each solution of "y" into x=1%2B2y to find "x"


Plug in y+=+2

x=1%2B2%282%29=1%2B4=5


So one set of solutions is x=5 and y=2 giving us the ordered pair (or point of intersection) (5,2)

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Plug in y+=+-6

x=1%2B2%28-6%29=1-12=-11


This means that another set of solutions is x=-11 and y=-6 giving us the ordered pair (or point of intersection) (-11,-6)


Here's a graph to visually confirm the answer:



Graph of x%5E2+-+3y%5E2+=+13 (red) and x+-+2y+=+1 (blue) with the intersection points (-11,-6) and (5,2) (black points)