SOLUTION: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not. 2x-5y=7, 3x+4y=2.

Question 1141180: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not.
2x-5y=7,
3x+4y=2.
Found 2 solutions by Theo, 4419875:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
start with:

2x-5y=7
3x+4y=2

multiply both sides of the first equation by 3 and multiply both sides of the second equation by 2 to get:

6x - 15y = 21
6x + 8y = 4

subtract the second equation from the first to get -23y = 17

solve for y to get y = -17/23

replace y with -17/23 in the first original equation to get 2x - 5y = 7 become 2x - 5 * -17/23 = 7.

multiply both sides of that equation by 23 to get 46x + 85 = 161

solve for x to get x = 76/46.

you have x = 76/46 and y = -17/23

replace x and y in both original equations with these values to get:

2x-5y=7 becomes 2 * 76/46 - 5 * -17/23 = 7.
multiply both sides of this equation by 46 to get 2 * 76 - 10 * -17 = 7 * 46.
simplify to get 152 + 170 = 322 which becomes 322 = 322 which is true.

3x+4y=2 becomes 3 * 76/46 + 4 * -17/23 = 2.
multiply both sides of this equation by 46 to get 3 * 76 + 8 * -17 = 2 * 46
simplify to get 228 - 136 = 92 which becomes 92 = 92 which is true.

both original equaitons are true when you replace x with 76/46 and y with -17/23, confirming the solution is correct.

your solution is x = 76/46 and y = -17/23.

x = 76/46 can be simplified to 38/23 while y remains simplified at -17/23.

those are your solutions.

Answer by 4419875(21) (Show Source): You can put this solution on YOUR website!
So what we're going to do here is elimination method:
1. Multiply the first one with 4 and 5 on the latter(since we wanted to cancel out
the y)
-------------------------------------

2. Add the second eq. from the first one
-------------------------------------

Hence, this would be like this:
=====================================

3. Divide both sides with 23 to isolate x and here we get:
---------------------------------------------------------

Now we have x and we could substitute it to any of the equations. From this point, I'll choose for the first eq.
4. Plug in x and subtract it from the left side to transfer it from the right side(applying subtraction property)
---------------------------------------------------------
4.1
4.2
4.3
From 4.3 Multiply 23 to 7 and subtract it with 76 and you get:

5. Divide both sides with -5 to isolate y and here we get:
---------------------------------------------------------

So our solution for the equations is ( , )
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