Question

If \(a = \frac{1}{3}b\) and \(b = 4c\), then in terms of c, \(a - b + c = ?\)

• A. c
• B. \(\frac{7}{3}c\)
• C. \(-\frac{5}{3}c\)
• D. \(-\frac{11}{3}c\)
• E.

Hint:

When performing a series of substitutions, be organized. Write down what each variable is in terms of your target variable (in this case, 'c') before plugging them into the final expression. This helps avoid confusion and errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are asked to evaluate an algebraic expression involving three variables (a, b, c) by expressing it solely in terms of one variable, c. This requires substituting the given relationships into the expression.
Step 2: Key Formula or Approach:
1. Express variable 'b' in terms of 'c' (this is already given).
2. Express variable 'a' in terms of 'c' by using the two given equations.
3. Substitute the expressions for 'a' and 'b' into the target expression \(a - b + c\).
4. Simplify the resulting expression.
Step 3: Detailed Explanation:
We are given: \[ b = 4c \] And: \[ a = \frac{1}{3}b \] First, substitute the expression for b into the equation for a to find a in terms of c: \[ a = \frac{1}{3}(4c) = \frac{4c}{3} \] Now we have both a and b in terms of c. We can substitute them into the expression \(a - b + c\): \[ a - b + c = \left(\frac{4c}{3}\right) - (4c) + c \] To combine these terms, we need a common denominator, which is 3. \[ = \frac{4c}{3} - \frac{12c}{3} + \frac{3c}{3} \] Now, combine the numerators: \[ = \frac{4c - 12c + 3c}{3} \] \[ = \frac{-8c + 3c}{3} \] \[ = \frac{-5c}{3} \] Step 4: Final Answer:
The expression \(a - b + c\) is equal to \(-\frac{5}{3}c\).