If
4a + 3b + c = 27
3a + b + 4c = 16
9a + 2b + 3c = 33
then 2a + 2b + 2c = ?
To find the value of 2a + 2b + 2c, we can first solve the given system of equations using the method of elimination or substitution. Let's use the method of elimination.
We have the following equations:
1) 4a + 3b + c = 27
2) 3a + b + 4c = 16
3) 9a + 2b + 3c = 33
First, we need to eliminate one variable. Let's eliminate the variable b.
Multiply Equation 2 by 3 and Equation 1 by -1:
-3a - b - 12c = -48 (Equation 1 multiplied by -1)
9a + 3b + 12c = 48 (Equation 2 multiplied by 3)
Add the equations together:
(9a + 3b + 12c) + (-3a - b - 12c) = 48 + (-48)
6a = 0
a = 0
Now, substitute the value of a into one of the original equations. Let's use Equation 1:
4(0) + 3b + c = 27
3b + c = 27
We can simplify this equation to:
c = 27 - 3b
Substituting a = 0 into Equation 3:
9(0) + 2b + 3c = 33
2b + 3c = 33
Replacing c in terms of b:
2b + 3(27 - 3b) = 33
2b + 81 - 9b = 33
-7b = -48
b = 48/7
Substituting b = 48/7 back into Equation 1 to find c:
3(48/7) + c = 27
144/7 + c = 27
c = 27 - 144/7
c = (189 - 144)/7
c = 45/7
Finally, substitute the values of a, b, and c into 2a + 2b + 2c:
2(0) + 2(48/7) + 2(45/7)
= 96/7 + 96/7 + 90/7
= (96 + 96 + 90)/7
= 282/7
Therefore, 2a + 2b + 2c equals 282/7.