Problem 96 - Beginning and Intermediate Algebra

Question

Write each expression without parentheses. $$ -(-z+5 w-9 y) $$

Step-by-Step Solution

Answer

z - 5w + 9y

Step 1: Identify the Expression

The given expression is \(-(-z + 5w - 9y)\)

Step 2: Distribute the Negative Sign

Apply the negative sign located outside the parentheses to each term within the parentheses. Recall that \(-(-a) = a\), so: \[-(-z + 5w - 9y) = z - 5w + 9y\]

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(z\), \(w\), \(y\)), and operations (such as addition, subtraction, multiplication, and division). They are used to represent relationships between quantities and can be simplified or evaluated given certain values for the variables. In this exercise, we work with an algebraic expression that involves parentheses and needs to be rewritten without them.

Negative Sign Distribution

One key concept to grasp for this type of exercise is the distribution of a negative sign across terms inside parentheses. To distribute a negative sign, multiply each term inside the parentheses by -1. This process changes the sign of each term.

For example, in the expression \(-(-z + 5w - 9y)\), we distribute the negative sign across each term within the parentheses:

• The negative outside and the negative \(z\) become positive \(z\): \(-(-z) = z\).
• The negative outside and the positive \(5w\) become negative \(5w\): \(-(5w) = -5w\).
• The negative outside and the negative \(9y\) become positive \(9y\): \(-(-9y) = 9y\).

So, the expression \(-(-z + 5w - 9y)\) simplifies to \(z - 5w + 9y\).

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves combining like terms and eliminating parentheses to make the expression more compact and easier to work with.

In our example, the initial expression is \(-(-z + 5w - 9y)\). By distributing the negative sign, we change each term inside the parentheses and remove the parentheses.

The process simplifies the expression to \(z - 5w + 9y\). This rewritten form is simpler and still has the same meaning as the original expression. Simplified expressions are more straightforward to use in further calculations, making problem-solving easier.