Problem 7 - Elementary and Intermediate Algebra

Question

Multiply both sides of the equation by \(2 .\) Do not solve. $$ 10=\frac{1}{2} b(b+5) $$

Step-by-Step Solution

Answer

The resulting equation is \( 20 = b(b+5) \).

Step 1: Identify the Equation

We are given the equation \( 10 = \frac{1}{2} b(b+5) \). Our task is to multiply both sides of the equation by 2. We do not need to solve the equation, just modify it according to the given instructions.

Step 2: Multiply Both Sides by 2

To eliminate the fraction, we multiply each term on both sides of the equation by 2. On the left side, \( 10 \times 2 = 20 \), and on the right side, \( 2 \times \frac{1}{2} b(b+5) = b(b+5) \).

Step 3: Write the Resulting Equation

After multiplying both sides by 2, the equation simplifies to \( 20 = b(b+5) \). This is the equation we obtain after performing the specified operation.

Key Concepts

Multiplying Equations

Multiplying equations involves applying the same operation to both sides of an equation. This technique aids in maintaining balance within the equation, which is vital to preserve equality. Indeed, the essence of any equation is that both sides are equal. Hence, any mathematical operation performed must be mirrored on both sides. This keeps the equation valid.

• In our original exercise, the instruction is to multiply by 2.
• It's imperative to apply the multiplication to both the left-hand side and right-hand side.

By doing so, the equation's form changes, but its integrity as a balanced equation remains intact.

Fraction Elimination

Fractions can make equations appear more complicated, but there's a neat way to simplify them: fraction elimination. This involves transforming fractions into whole numbers, making the arithmetic more straightforward. In the context of linear equations, this often requires multiplying by the denominator.

• For example, in the equation from the exercise, the right side is given as \( \frac{1}{2} b(b+5) \).
• By multiplying the entire equation by 2, we are essentially getting rid of the fraction. This action simplifies the expression to \( b(b+5) \).

The disappearance of the fraction allows us to work with whole numbers, which are generally easier to manipulate algebraically.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in mathematics, involving rearranging and simplifying equations to make them easier to work with. It includes actions like distributing, combining like terms, and factoring expressions.
In the given exercise, after multiplying both sides of the equation by 2, the expression simplifies to \( 20 = b(b+5) \). This straightforward expression allows for easier subsequent algebraic operations.

• Each manipulation step should aim to maintain the equation's balance.
• Through algebraic manipulation, we often aim to isolate a variable or simplify complex expressions.

This technique is essential as it forms the foundation for solving more complex algebraic equations effectively.