solve by elimination
5r-6s=18
6r-5s=46
The solution is
There are infinitely many solutions
there is no solution
There is a unique solution.
5r-6s=18 ...(1)
6r-5s=46 ...(2)
6(1)
30r-36s=108...(1a)
30r-25s=230...(2a)
(2a)-(1a)
0r+11s = 122
s=122/11
r=(18+6*122/11)/5
= 186/11
Back-substitute to make sure the answers are correct.
So is there no solution to this?
To solve the system of equations using elimination, the goal is to eliminate one of the variables by manipulating the equations.
Given the system of equations:
1) 5r - 6s = 18
2) 6r - 5s = 46
To eliminate a variable, you need to multiply one or both equations by appropriate numbers so that the coefficients of either r or s become the same (but with opposite signs) in both equations.
In this case, we can multiply equation 1 by 6 and equation 2 by 5 to make the coefficients of s the same but with opposite signs:
3) 30r - 36s = 108
4) 30r - 25s = 230
Now, subtract equation 4 from equation 3:
(30r - 36s) - (30r - 25s) = 108 - 230
30r - 36s - 30r + 25s = -122
-11s = -122
Divide both sides of the equation by -11 to solve for s:
s = -122 / -11
s = 11
Substituting the value of s back into equation 1:
5r - 6(11) = 18
5r - 66 = 18
5r = 18 + 66
5r = 84
r = 84 / 5
r = 16.8
Therefore, the solution to the system of equations is r = 16.8 and s = 11.
Since there is a unique solution for both r and s, there are no infinite solutions and no contradiction. Hence, the answer is that there is a unique solution, and there is no case where there is no solution.