Question

If $a - 6b + 6c = 4$ and $6a + 3b - 3c = 50$, where $a, b$ and $c$ are real numbers, the value of $2a + 3b - 3c$ is:

• A. \(18\)
• B. \(20\)
• C. \(15\)
• D. \(14\)

Hint:

When several linear expressions repeat with the same structure (like $3b - 3c$ here), use substitution to reduce the system to two variables. This often simplifies the equations dramatically.

Answer 1

The Correct Option is A

We are given: \[ a - 6b + 6c = 4, \] \[ 6a + 3b - 3c = 50. \] We want: \[ E = 2a + 3b - 3c. \] Step 1: Introduce a substitution. Notice that both the second given equation and the target expression contain the term: \[ 3b - 3c. \] Let: \[ X = 3b - 3c. \] Use this substitution in the equations. \underline{Rewrite the first equation:} \[ a - 6b + 6c = a - 2(3b - 3c) = a - 2X = 4. \tag{i} \] \underline{Rewrite the second equation:} \[ 6a + (3b - 3c) = 6a + X = 50. \tag{ii} \] \underline{Rewrite the target expression:} \[ E = 2a + (3b - 3c) = 2a + X. \tag{iii} \]
Step 2: Solve equations (i) and (ii). From (ii): \[ X = 50 - 6a. \] Substitute into (i): \[ a - 2(50 - 6a) = 4, \] \[ a - 100 + 12a = 4, \] \[ 13a - 100 = 4, \] \[ 13a = 104, \] \[ a = 8. \] Now compute \(X\): \[ X = 50 - 6(8) = 50 - 48 = 2. \]
Step 3: Evaluate the target expression. From (iii): \[ E = 2a + X = 2(8) + 2 = 16 + 2 = 18. \] \[ \boxed{18} \]