If
4a + 3b + c = 27
3a + b + 4c = 16
9a + 2b + 3c = 33
then 2a + 2b + 2c = ?
Why don't you just solve the matrix on your calc, or with row reduction, then add a+b+c?
Or to get it algebraically, call the equations P, Q and R respectively, so
P: 4a + 3b + c = 27
Q: 3a + b + 4c = 16
R: 9a + 2b + 3c = 33
Solve these for b first, since the multiples are easy:
R-2Q = 3a - 5c = 1
P-3Q = -5a - 11c = -21
so
5(R-2Q) = 15a - 25c = 5
3(P-3Q) = -15a - 33c = -63
Add these together and you get -58c = 58, so c=1, therefore a=2. Feed a and c into any of the original equations and you get b=6. So a+b+c=9, and twice this is 18.
Correction: -58c=-58, so c=1.
To solve this system of equations, we can use the method of substitution or elimination. Here, I will explain the method of substitution.
Step 1: Solve one equation for one variable.
Let's solve the first equation for a:
4a + 3b + c = 27
4a = 27 - 3b - c
a = (27 - 3b - c) / 4
Step 2: Substitute the value of a into the remaining two equations.
Substitute (27 - 3b - c) / 4 for a in the second and third equations:
3((27 - 3b - c) / 4) + b + 4c = 16 (Equation 2)
9((27 - 3b - c) / 4) + 2b + 3c = 33 (Equation 3)
Now, simplify the equations:
Equation 2:
(81 - 9b - 3c + 4b + 16c) / 4 + b + 4c = 16
81 - 9b - 3c + 4b + 16c + 4b + 16c = 64 + 36
-5b + 29c = -11 (Equation 4)
Equation 3:
(243 - 27b - 9c + 8b + 33c) / 4 + 2b + 3c = 33
243 - 27b - 9c + 8b + 33c + 8b + 12c = 132
-19b + 36c = -111 (Equation 5)
Step 3: Solve the system of linear equations.
Now, we need to solve equations 4 and 5 simultaneously.
Multiply equation 4 by 19 and equation 5 by 5 to eliminate b.
-95b + 551c = -209 (Equation 6)
-95b + 180c = -555 (Equation 7)
Subtract equation 7 from equation 6 to eliminate b:
371c = 346
c = 346 / 371
c ≈ 0.933
Substitute the value of c back into equation 4 or 5:
-5b + 29(0.933) = -11
-5b + 26.957 = -11
-5b = -11 - 26.957
-5b = -37.957
b = (-37.957) / -5
b ≈ 7.5914
Step 4: Substitute the values of b and c into any of the original equations to find the value of a.
Let's substitute b and c into the first equation:
4a + 3b + c = 27
4a + 3(7.5914) + 0.933 = 27
4a + 22.7742 + 0.933 = 27
4a + 23.7072 ≈ 27
4a ≈ 27 - 23.7072
4a ≈ 3.2928
a ≈ 3.2928 / 4
a ≈ 0.8232
Step 5: Find the value of 2a + 2b + 2c.
2a + 2b + 2c = 2(0.8232) + 2(7.5914) + 2(0.933)
2a + 2b + 2c ≈ 1.6464 + 15.1828 + 1.866
2a + 2b + 2c ≈ 18.6952
Therefore, 2a + 2b + 2c is approximately equal to 18.6952.