Problem 31 - Algebra 1

Question

Simplify the variable expression. $$(7)(-x)$$

Step-by-Step Solution

Answer

-7x

Step 1: Identify multiplication components

The given expression is \( (7)(-x) \), which involves two components: constant \( 7 \) and variable \( -x \).

Step 2: Carry out multiplication

Multiply the two components to simplify the expression. \( 7 \) times \( -x \) equals \( -7x \), as whenever a positive number is multiplied by a negative number, the result is a negative number.

Key Concepts

Variable Expression Multiplication

Multiplication is a fundamental operation in algebra, especially when it involves variables. When we talk about variable expression multiplication, we're looking at the process of combining constants and variables to simplify an expression. It's a bit like cooking with a recipe; each ingredient (constant or variable) has a specific role and quantity.

For example, consider the numeral as a whole number ingredient and the variable as a special spice. You multiply these two to create a new product, your simplified expression. In our textbook example, the expression is simplified by multiplying a constant 7 with a variable -x. The process is straightforward, just like regular multiplication, but we must not forget to apply the rules of algebra, particularly when dealing with positive and negative numbers.

• Identify constants and variables
• Follow the sign rules for multiplication
• Multiply the numeral by the variable to get a simplified expression

Understanding this concept paves the way for more complex algebraic manipulations and is essential for mastering algebra.

Negative Numbers in Algebra

Algebra often feels like navigation, especially when it comes to understanding the role of negative numbers. Negative numbers, those less than zero, play a crucial part in understanding algebraic expressions. They are like the opposite direction in our navigation, and they can change the destination of our mathematical journey.

One important rule to remember is that multiplying a positive number by a negative number yields a negative result, similar to how two wrongs don't make a right. When faced with an expression like \((7)(-x)\), the result, \(-7x\), reflects this principle. Dealing with negative numbers requires attention to detail, as overlooking the sign can lead to errors.

• Remember that a negative times a positive equals a negative
• Keep track of negative signs to avoid mistakes
• Practice with different expressions to become comfortable with negative numbers

Grasping this concept will significantly enhance a student's confidence in solving algebraic problems.

Multiplication with Variables

When working with multiplication that involves variables, it's like entering a world where letters and numbers coexist harmoniously. Variables represent unknown quantities, and they're denoted by letters like x, y, or z. Multiplying them with constants or other variables follows the same rules as numeral multiplication, but with an added layer of abstraction.

In the case of the example problem, \((7)(-x)\), our goal is to combine the constant 7 (which we know) with the variable -x (which stands for an unknown number). The multiplication rule states that the two can be combined by multiplying their values, leading to \(-7x\). This process lays the groundwork for more advanced algebra, such as working with polynomial expressions.

• Understand the role of variables as unknowns
• Apply standard multiplication rules to variables and constants
• Practice with simple and complex expressions to build algebraic fluency

By mastering multiplication with variables, you'll unlock the ability to solve for the unknowns in any algebraic equation.