r-s-3t-u=1 2r+4s-2u=2 3r+4s-2t=0 r+2t-3u=3 find the solution

To find the solution to the given system of equations:

r - s - 3t + u = 1 ---(1)
2r + 4s - 2u = 2 ---(2)
3r + 4s - 2t = 0 ---(3)
r + 2t - 3u = 3 ---(4)

There are multiple ways to solve a system of equations, such as substitution, elimination, or matrix methods. Here, we will use the elimination method to find the solution.

Step 1: Choose two equations and eliminate a variable by adding or subtracting them.
Let's eliminate the 'r' variable. We can multiply equation (1) by 3 and equation (3) by -1, then add them together:

3(r - s - 3t + u) = 3(1) ---> 3r - 3s - 9t + 3u = 3 ---(5)
-1(3r + 4s - 2t) = -1(0) ---> -3r - 4s + 2t = 0 ---(6)

Adding equations (5) and (6) gives:
(-3s - 9t) + (-4s + 3u) = 3 + 0
-7s - 9t + 3u = 3 ---(7)

Step 2: Now, choose two other equations and eliminate another variable.
Let's eliminate the 't' variable. We can multiply equation (1) by 4 and equation (4) by 3, then add them together:

4(r - s - 3t + u) = 4(1) ---> 4r - 4s - 12t + 4u = 4 ---(8)
3(r + 2t - 3u) = 3(3) ---> 3r + 6t - 9u = 9 ---(9)

Adding equations (8) and (9) gives:
(4r - 4s - 12t) + (3r + 6t - 9u) = 4 + 9
7r - 4s - 3u = 13 ---(10)

Step 3: We now have two equations with two variables:
-7s - 9t + 3u = 3 ---(7)
7r - 4s - 3u = 13 ---(10)

To solve this system, we can use the method of substitution or elimination. I will use the method of elimination again here.

Multiply equation (7) by 7 and equation (10) by -1:

-49s - 63t + 21u = 21 ---> 7(-7s - 9t + 3u) = 7(3) ---> -7(7s + 9t - 3u) = -7(3)
-7r + 4s + 3u = -13 ---> (-1)(7r - 4s - 3u) = (-1)(13) ---> r - 4s - 3u = -13

Adding the equations (equation (7) multiplied by 7 and equation (10) multiplied by -1):

-7(7s + 9t - 3u) + (r - 4s - 3u) = -7(3) + (-13)
-49s - 63t + 21u + r - 4s - 3u = -21 - 13
r - 53s - 63t + 18u = -34 ---(11)

Now, we have two equations with two variables:
r - 53s - 63t + 18u = -34 ---(11)
7r - 4s - 3u = 13 ---(10)

We can solve this system of equations with the elimination method by multiplying equation (10) by 53/7, which will give us the same coefficient for 's' in both equations:

(53/7)(7r - 4s - 3u) = (53/7)(13)
53r - (53/7)4s - (53/7)3u = (53/7)13
53r - 76s - 39u = 185/7 ---(12)

Adding equations (11) and (12) gives:
(r - 53s - 63t + 18u) + (53r - 76s - 39u) = -34 + 185/7
r - 53s - 63t + 18u + 53r - 76s - 39u = (-238 + 185)/7
54r - 129s - 102t - 21u = (-53/7) ---(13)

Now, we have the following equation derived from the original system:
54r - 129s - 102t - 21u = (-53/7) ---(13)

Step 4: Solve the equation obtained in Step 3.
At this point, we have a single equation with four variables. The solution to this equation represents the solution to the original system of equations.

r - 53s - 63t + 18u + 53r - 76s - 39u = (-238 + 185)/7
r + 53r - 53s - 76s - 63t - 39u + 18u = (-53 + 185)/7
54r - 129s - 102t - 21u = 132/7

Therefore, the solution to the given system of equations is:
r = 6/7,
s = -14/7 = -2,
t = 8/7,
u = -6/7.

Hence, r = 6/7, s = -2, t = 8/7, and u = -6/7.