Understanding Solutions in Algebra: Substituting $ a = 2b $ into the Expression for $ E $

When solving algebraic expressions, substitution is a powerful and frequently used technique. One common application involves expressing one variable in terms of another and substituting into an equation—often leading to simplified or more insightful forms. In this article, we’ll explore a key algebraic scenario where $ a = 2b $ is substituted into an expression for $ E $, explaining step-by-step how such substitution relies on defined relationships and transforms equations into usable forms.

Understanding the Context

The Expression for $ E $

Suppose we are given a quantity $ E $ defined as:

$$
E = a^2 + b
$$

This expression combines two variables, $ a $ and $ b $, which may share a functional relationship. For many problems in algebra and applied mathematics, relationships between variables aren’t just defined arbitrarily—they are given by equations that allow substitution.

Solution: Given $ a = 2b $, substitute into the expression for $ E $:

Key Insights

Applying the Relationship $ a = 2b $

We are told to substitute $ a = 2b $ into the expression for $ E $. Substitution means replacing the variable $ a $ with $ 2b $ throughout the expression wherever it appears.

Starting with:

$$
E = a^2 + b
$$

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Solution: Group and complete the square for $ x $ and $ y $.
For $ x $: $ 9(x^2 - 4x) = 9[(x - 2)^2 - 4] $.
For $ y $: $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $.

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Final Thoughts

Substitute $ a = 2b $:

$$
E = (2b)^2 + b
$$

Simplifying the Result

Now simplify the expression:

$$
E = (2b)^2 + b = 4b^2 + b
$$

Thus, by substituting $ a = 2b $ into the original expression, the new form of $ E $ becomes:

$$
E = 4b^2 + b
$$

This simplified expression depends solely on $ b $, making it easier to analyze, differentiate, integrate, or solve depending on context.