Problem 2 - Technical Mathematics with Calculus

Question

Remove parentheses and simplify. $$-(b-c)$$

Step-by-Step Solution

Answer

-(b - c) simplifies to -b + c.

Step 1: Apply the distributive property

To remove the parentheses, multiply the term outside the parentheses by each term inside. Here, you should multiply -1 by both b and -c, which gives -1 * b + -1 * (-c). Remember that multiplying by -1 inverts the sign of the term.

Step 2: Simplify the expression

After distributing the negative sign, the expression simplifies to -b + c, because -1 multiplied by b is -b and -1 multiplied by -c is +c.

Key Concepts

Distributive Property

Understanding the distributive property is key to simplifying algebraic expressions. It allows us to remove parentheses by distributing a single term outside the parentheses across each term inside the parentheses. In mathematical terms, the distributive property is explained by the formula: \(a(b+c) = ab + ac\).
For example, if we have 3(x + y), we apply the distributive property by multiplying 3 times x and 3 times y, which results in 3x + 3y.

When we look at a negative sign outside parentheses, like in our exercise \( -(b-c) \), we interpret it as \( -1 \) multiplied by each term inside the parentheses. This is a crucial step in understanding how to change the signs of terms inside the parentheses as per the distributive rule. By following this method, we ensure that every term is accounted for correctly in the simplification process.

Removing Parentheses

Removing parentheses from an algebraic expression is a straightforward process once you understand the distributive property. It's like untangling a knot; you must effectively apply the distributive property or 'distribute the multiplier', and the expression will unravel itself.

When dealing with the expression \( -(b - c) \), you can think of parentheses as holding the terms together. To remove these parentheses, we need to ensure that the sign or number directly in front of the parentheses is applied to each term within. This frees the terms from their confinement and allows them to stand independently within the expression. Always ensure that you carry the sign in front of the parentheses through to each term within it to avoid any sign errors.

Inverting Signs

Inverting signs is an essential step when dealing with negative signs outside parentheses, especially because it alters the value of the terms inside. If a negative sign (or a subtraction sign) appears before the parentheses, we invert the sign of each term within the parentheses as we remove them. This is directly tied to the rule that \( -1 \) times any positive term becomes negative, and \( -1 \) times a negative term becomes positive.

For instance, when simplifying \( -1(b - c) \), which is the same as \( -(b - c) \), the \( -1 \) outside flips the signs inside: \( -1 \) times \( b \) becomes \( -b \) and \( -1 \) times \( -c \) becomes \( +c \). This sign inversion is a common source of mistakes, and it is paramount to handle it meticulously to arrive at the correct simplified expression.