Question

\(m = 4x + 4y\), \(x \neq -y\)
 

Column A	Column B
\(\frac{2m}{x+y}\)	8

Hint:

Whenever you see an algebraic expression to be simplified, always look for common factors that can be canceled out. The condition \(x \neq -y\) is a strong hint that the term \((x+y)\) will be in the denominator and needs to be canceled.

Answer 1

Step 1: Understanding the Concept:
This is an algebraic manipulation problem. We need to substitute the given expression for \(m\) into the expression in Column A and simplify it.
Step 2: Key Formula or Approach:
The key steps are substitution and factorization. We will factor a common term from the expression for \(m\) to allow for simplification.
Step 3: Detailed Explanation:
For Column A:
The expression is \(\frac{2m}{x+y}\).
We are given \(m = 4x+4y\). Substitute this into the expression in Column A:
\[ \frac{2(4x+4y)}{x+y} \] Now, look at the term in the parentheses, \(4x+4y\). We can factor out a common factor of 4:
\[ 4x+4y = 4(x+y) \] Substitute this factored form back into the expression:
\[ \frac{2 \times 4(x+y)}{x+y} \] \[ \frac{8(x+y)}{x+y} \] We are given the condition that \(x \neq -y\), which means \(x+y \neq 0\). Since the denominator is not zero, we can safely cancel the \((x+y)\) term from the numerator and the denominator.
\[ \frac{8\cancel{(x+y)}}{\cancel{(x+y)}} = 8 \] The value of Column A is 8.
Step 4: Final Answer:
Comparing the two quantities:
Column A = 8
Column B = 8
The two quantities are equal.