Problem 6 - Calculus for Biology and Medicine

Question

$$ \text { Show that } 2(A+B)=2 A+2 B \text { . } $$

Step-by-Step Solution

Answer

The equation holds true by the distributive property: \(2(A+B)=2A+2B\).

Step 1: Understand the Problem

We need to show that the left side of the equation, \(2(A+B)\), equals the right side, \(2A + 2B\). This involves using the distributive property to expand the left side.

Step 2: Apply the Distributive Property

The distributive property in algebra states that \(a(b+c) = ab + ac\). Applying this property to \(2(A+B)\), we distribute the 2 across the sum inside the parentheses: \(2(A+B) = 2 \cdot A + 2 \cdot B\).

Step 3: Simplify Both Sides

After distribution, we simplify the expression: \(2A + 2B\) on the left side. This matches exactly with the right side of the equation, which is \(2A + 2B\).

Step 4: Conclusion

Since both sides of the equation simplify to \(2A + 2B\), we confirm that the initial equation \(2(A+B)=2A+2B\) holds true.

Key Concepts

Algebra: Understanding Distributive Property

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. One of the fundamental properties in algebra is the distributive property. This property helps in simplifying expressions and solving equations. It states that multiplying a number by a sum is the same as doing each multiplication separately. In simpler terms, it allows you to "distribute" the multiplication over addition or subtraction in a bracket.
For example, with the expression \( a(b+c) \), you can apply the distributive property to get \( ab + ac \). This step is crucial in simplifying equations as it helps break down complex expressions into more manageable parts.

Equation Simplification: Breaking Down Expressions

Equation simplification is the process of making an equation easier to understand or solve. This involves combining like terms, distributing constants, and eliminating unnecessary operations. The primary goal is to transform a complex equation into a simpler, equivalent one that is easier to work with.
In the given exercise, the equation \( 2(A+B) \) is simplified using the distributive property. By distributing the 2 across \( (A+B) \), the expression becomes \( 2A + 2B \). Simplification makes both sides of the equation identical and clearer to interpret. It also leads to a quicker identification of solutions or proofs for equalities.

Mathematical Proof: Demonstrating Equality

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. It involves a series of deductive steps based on axioms, definitions, and previously established theorems or results. Proofs are essential in mathematics to establish the legitimacy of equations and relationships between numbers or expressions.
To show that \( 2(A+B) = 2A + 2B \), we used a proof that relies on applying the distributive property. By expanding and simplifying the left side of the equation, we arrive at the expression \( 2A + 2B \), which matches the right side. This verifies the equality of both sides, completing the proof. Proofs ensure that mathematical conclusions are sound and indisputable, reinforcing the principles of logic in mathematics.