Let a, b, c such that `(1)/((1-x)(1-2x)(1-3x))=(a)/(1-x)+(b)/(1-2x)+(c)/(1-3x), a/1+b/3+c/5=`

Options

A. `1/15`
B. `1/6`
C. `1/5`
D. `1/3`

Answer

To solve the equation \(\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}\), we will follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ \frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x} \]

### Step 2: Clear the denominators
Multiply both sides by \((1-x)(1-2x)(1-3x)\) to eliminate the denominators:
\[ 1 = a(1-2x)(1-3x) + b(1-x)(1-3x) + c(1-x)(1-2x) \]

### Step 3: Expand the right-hand side
Now, we will expand the right-hand side:
1. \(a(1-2x)(1-3x) = a(1 - 5x + 6x^2)\)
2. \(b(1-x)(1-3x) = b(1 - 4x + 3x^2)\)
3. \(c(1-x)(1-2x) = c(1 - 3x + 2x^2)\)

Combining these, we have:
\[ 1 = a(1 - 5x + 6x^2) + b(1 - 4x + 3x^2) + c(1 - 3x + 2x^2) \]

### Step 4: Collect like terms
Now we collect the coefficients of \(1\), \(x\), and \(x^2\):
\[ 1 = (a + b + c) + (-5a - 4b - 3c)x + (6a + 3b + 2c)x^2 \]

### Step 5: Set up the system of equations
From the equation above, we can equate coefficients:
1. Constant term: \(a + b + c = 1\) (Equation 1)
2. Coefficient of \(x\): \(-5a - 4b - 3c = 0\) (Equation 2)
3. Coefficient of \(x^2\): \(6a + 3b + 2c = 0\) (Equation 3)

### Step 6: Solve the system of equations
From Equation 1, we can express \(c\) in terms of \(a\) and \(b\):
\[ c = 1 - a - b \]

Substituting \(c\) into Equations 2 and 3:
1. \(-5a - 4b - 3(1 - a - b) = 0\)
\[ -5a - 4b - 3 + 3a + 3b = 0 \implies -2a - b - 3 = 0 \implies b = -2a - 3 \quad (Equation 4) \]

2. \(6a + 3b + 2(1 - a - b) = 0\)
\[ 6a + 3b + 2 - 2a - 2b = 0 \implies 4a + b + 2 = 0 \implies b = -4a - 2 \quad (Equation 5) \]

### Step 7: Equate and solve for \(a\)
From Equations 4 and 5:
\[ -2a - 3 = -4a - 2 \]
Solving this gives:
\[ 2a = 1 \implies a = \frac{1}{2} \]

### Step 8: Find \(b\) and \(c\)
Using \(a = \frac{1}{2}\) in Equation 4:
\[ b = -2\left(\frac{1}{2}\right) - 3 = -1 - 3 = -4 \]

Using \(a\) and \(b\) in Equation 1:
\[ c = 1 - \frac{1}{2} + 4 = 1 - \frac{1}{2} + 4 = \frac{1}{2} + 4 = \frac{9}{2} \]

### Step 9: Calculate \( \frac{a}{1} + \frac{b}{3} + \frac{c}{5} \)
Now we compute:
\[ \frac{a}{1} + \frac{b}{3} + \frac{c}{5} = \frac{1/2}{1} + \frac{-4}{3} + \frac{9/2}{5} \]
Calculating each term:
- \( \frac{1}{2} = 0.5 \)
- \( \frac{-4}{3} \approx -1.3333 \)
- \( \frac{9/2}{5} = \frac{9}{10} = 0.9 \)

Adding these:
\[ 0.5 - 1.3333 + 0.9 = 0.5 + 0.9 - 1.3333 = 1.4 - 1.3333 = 0.0667 \approx \frac{1}{15} \]

Thus, the final answer is:
\[ \frac{a}{1} + \frac{b}{3} + \frac{c}{5} = \frac{1}{15} \]