The given system of equations is:
s/2 + 5t = 3 + 3s - 6 ...(1)
3t - 6s = 9 + 2s + 7t ...(2)
Let's solve this system.
First, let's simplify equation (1):
s/2 + 5t = 3 + 3s - 6
Multiply both sides of the equation by 2 to eliminate the fraction:
s + 10t = 6 + 6s - 12
Rearranging the terms:
10t - s = 6s - s - 12 + 6
10t - s = 5s - 6 ...(3)
Now, let's simplify equation (2):
3t - 6s = 9 + 2s + 7t
Rearranging the terms:
3t - 7t - 6s - 2s = 9
-4t - 8s = 9 ...(4)
Next, let's solve equations (3) and (4) simultaneously.
Multiply equation (3) by 4 and equation (4) by -5, then add them:
4(10t - s) + (-5)(-4t - 8s) = 4(5s - 6) + (-5)(9)
40t - 4s + 20t + 40s = 20s - 24 - 45
Collect like terms:
60t + 36s = 20s - 69
Rearranging the terms:
60t + 36s - 20s = -69
Simplifying:
60t + 16s = -69 ...(5)
Now, we have two equations: (4) and (5).
Let's eliminate s by multiplying equation (5) by -8 and adding it to equation (4):
-8(60t + 16s) + (-4t - 8s) = -8(-69)
-480t - 128s - 4t - 8s = 552
Collect like terms:
-484t - 136s = 552
Divide through by -4 to simplify:
121t + 34s = -138 ...(6)
Now we have two equations: (4) and (6).
Let's eliminate t by multiplying equation (6) by -3 and adding it to equation (4):
-3(121t + 34s) + (-4t - 8s) = -3(-138)
-363t - 102s - 4t - 8s = 414
Collect like terms:
-367t - 110s = 414
Now, we have a new linear equation in terms of s and t.
From equation (4):
-4t - 8s = 9
From equation (6):
-367t - 110s = 414
Solving these two equations simultaneously will give us the values of s and t.
We multiply equation (4) by 367 and equation (6) by -4, then add:
367(-4t - 8s) + (-4)(-367t - 110s) = 367(9) + (-4)(414)
-1468t - 2936s + 1468t + 440s = 3303 - 1656
Collect like terms:
-2496s = 1647
Dividing through by -2496:
s = -1647/2496
Next, substitute this value of s back into equation (4) to find t:
-4t - 8(-1647/2496) = 9
Simplifying:
-4t + 3294/2496 = 9
-4t = 9 - 3294/2496
-4t = (2232 - 3294)/2496
-4t = -1062/2496
Dividing through by -4:
t = (-1062/2496)/(-4)
t = 1062/9984
Therefore, the ordered pair (s, t) that satisfies the system of equations is:
(s, t) = (-1647/2496, 1062/9984)
Find the ordered pair (s,t) that satisfies the system
s/2 + 5t = 3 + 3s - 6
3t - 6s = 9 + 2s + 7t
1 answer
1 answer