SOLUTION: Trying to find a simple method of sovling the below linear equations for now and in the future. I thought I had it , when I set one variable to zero and solved for the remaining

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Question 891212: Trying to find a simple method of sovling the below linear equations for now and in the future.
I thought I had it , when I set one variable to zero and solved for the remaining variable, then reversed the process. finally replacing the variable with the solutions. left me with mixed solutions.
Need help solving the linear system of equations:

2.
2x-6y=7, 5x+2y=10
A. Unique solution
(8,3)
B. Unique solution
(37/17, -15/34),
C. Infinite many solutions
t,6t+1)
D. No solution

3.
x-16y=9, 1/4x +4y= 8
A. Unique solution
(9,4)
B. Unique solution
(2, -4)
C. Infinite many solutions
t,8t+9)
D. No solution

4.
3x - 2y=7, 9x 6y = 14
A. Unique solution
(6,-4)
B. Unique solution
(8, 5),
C. Infinite many solutions
t,8t+5)
D. No solution

Thanks
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
You are not justified in setting a variable equal to zero. You might do that if you are looking for axis intercepts for one linear equation. You are not interested in such a solution for solving simultaneous linear equations.

Number 2 as an example:
2x-6y=7, 5x+2y=10.

These will intersect since slopes are 1/3 and -5/2. Not equal slopes, and not negative reciprocals.

Substitution Method,
2x=7%2B6y
x=%287%2B6y%29%2F2;
5%287%2B6y%29%2F2%2B2y=10
5%287%2B6y%29%2B4y=20
35%2B30y%2B4y=20
34y=20-35
34y=-15
highlight%28y=-15%2F34%29.
-
x=%287%2B6%28-15%2F34%29%29%2F2
x=7%2F2-3%2815%2F34%29
7%2F2-45%2F34
%287%2A17-45%29%2F34
x=74%2F34------------(Alerted to mistake: now fixed; a digit "7" was missing)
highlight%28x=37%2F17%29
-
Solution is x=37%2F17 and y=-15%2F34.


--
Possibly I made a mistake in one part of the solution. Look through my steps and maybe you will find the (possible ) mistake.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Trying to find a simple method of sovling the below linear equations for now and in the future.
I thought I had it , when I set one variable to zero and solved for the remaining variable, then reversed the process. finally replacing the variable with the solutions. left me with mixed solutions.
Need help solving the linear system of equations:

2.
2x-6y=7, 5x+2y=10
A. Unique solution
(8,3)
B. Unique solution
(37/17, -15/34),
C. Infinite many solutions
t,6t+1)
D. No solution

3.
x-16y=9, 1/4x +4y= 8
A. Unique solution
(9,4)
B. Unique solution
(2, -4)
C. Infinite many solutions
t,8t+9)
D. No solution

4.
3x - 2y=7, 9x 6y = 14
A. Unique solution
(6,-4)
B. Unique solution
(8, 5),
C. Infinite many solutions
t,8t+5)
D. No solution

Thanks


system+%282x+-+6y+=+7_____eq+%28i%29%2C+5x+%2B+2y+=+10____eq+%28ii%29%29



17x = 37 ------ Adding eqs (iii) & (i)

highlight_green%28x+=+37%2F17%29



2%2837%2F17%29+-+6y+=+7 ----- Substituting 37%2F17 for x in eq (i)

74%2F17+-+6y+=+7

74%2F17+-+6y+=+119%2F17

-+6y+=+119%2F17+-+74%2F17

-+6y+=+45%2F17

y+=+%2845%2F17%29%2F-+6

y+=+%2845%2F17%29+%2A+%281%2F-6%29

y+=+-+45%2F102

highlight_green%28y+=+-+15%2F34%29 ----- Reducing fraction to its lowest term