Problem 32 - Algebra A Combined Function

Question

Simplify each expression. Use the distributive property to remove any parentheses. $$ -(y+5 z-7) $$

Step-by-Step Solution

Answer

The simplified expression is \[-y - 5z + 7\].

Step 1: Understand the expression

We are given the expression \[-(y+5z-7)\],which means we need to distribute the negative sign across the terms inside the parentheses.

Step 2: Distribute the negative sign

Apply the distributive property by multiplying each term inside the parentheses by \(-1\).- \(-(y) = -y\)- \(-(+5z) = -5z\)- \(-(-7) = +7\)

Step 3: Write the simplified expression

After distributing the negative sign, combine the resulting terms:\[-y - 5z + 7\].

Key Concepts

Expression Simplification

Simplifying an expression involves making it easier to work with by combining like terms and removing any unnecessary parentheses.
It's like cleaning up and organizing your thoughts in math form. When simplifying, one of our main goals is to make sure everything is as straightforward as possible.
This means ensuring that each term in the expression looks clean and free of any extra symbols or brackets that aren’t needed anymore. Here's a simple way to approach it:

• First, resolve operations inside parentheses. Use properties like the distributive property.
• Next, combine like terms. These are terms that have the same variable part and can be added or subtracted directly.
• Simplify further if possible, ensuring the expression is in its simplest form.

Understanding how to simplify an expression lays a solid foundation for solving more complicated algebraic problems later. Knowing the methods and steps to get rid of unnecessary parts is very useful.

Algebraic Expressions

Algebraic expressions are like the building blocks of algebra, composed of numbers, variables, and operation signs.
They represent mathematical ideas in a concise form. Think of a simple expression like \(3x + 4\). Here, 3 is a coefficient (the number in front of the variable), \(x\) is a variable that can represent many possible numbers, and 4 is a constant term.
Algebraic expressions can be used to represent all kinds of scenarios, from simple to complex.
Knowing how to manipulate these expressions by performing operations - like addition, subtraction, multiplication, and the application of properties like the distributive property - allows you to solve algebraic equations.

• They can stand for consistent values when the variables are known.
• Or serve as formulas to find unknowns in equations.
• Learning how to manage expressions aids in problem-solving processes across mathematics.

Getting comfortable with algebraic expressions means having them do the heavy lifting in more complex mathematical investigations later on.

Negative Sign Distribution

Distribution, especially of a negative sign, is a crucial part of working with expressions. When a negative sign appears in front of a parentheses group, it indicates that each term inside needs to be multiplied by \(-1\). Using the guide expression \(- (y + 5z - 7)\), it’s important to distribute the negative sign correctly for simplification:

• Multiply \(-1\) through each term: the coefficients and the signs change.
• Thus, the expression becomes \(-y - 5z + 7\).

Think of this as a way to "flip" each term’s sign inside the grouping. Understanding this process helps ensure you accurately simplify any mathematical statement or equation.
This technique doesn’t just stop at standalone expressions but is vital for solving larger equations as well. Syntax errors here can cause broader mistakes in answers and calculations.