Problem 63 - Algebra 1

Question

Simplify the expression. $$-4 x \div(-4)$$

Step-by-Step Solution

Answer

The expression \( -4x \div -4 \) simplifies to \( x \)}.

Step 1: Identify the constants that will be divided

Here, the expression \( -4x \div -4 \) involves two constants: \( -4 \) (the coefficient of \( x \)) and \( -4 \) (the constant term to be divided by).

Step 2: Apply division operation on constants

Divide the coefficient of \( x \) which is \( -4 \) by the constant \( -4 \). This is equivalent to solving \( -4 \div -4 \).

Step 3: Simplify the division

According to the rule, the division of two negative numbers is a positive number. As such, \( -4 \div -4 \) simplifies to 1.

Step 4: Write the simplified expression

After carrying out the division, the simplified expression is \( 1x \) or simply \( x \)

Key Concepts

Algebraic Division

When simplifying algebraic expressions, division plays a fundamental role, particularly when it involves coefficients and variables. Dividing algebraic terms requires understanding that each term consists of a coefficient (a numerical part) and, often, a variable (an alphabetic part). In the presented exercise, the expression (-4x (-4) is focused solely on the division of coefficients, as both terms involve the same variable.
When dividing algebraic expressions, we must also consider the laws of arithmetic, just as we would with ordinary numbers. If a variable is present, it's vital to ensure that we divide its coefficients while keeping the variable intact. For instance, if the expression had been (-4x (-2), the result would still involve the variable x, yielding 2x after completing the division. Simplifying algebraic terms through division is an essential step in solving algebra-related problems. Mastery of this concept will drastically improve the ability to tackle algebraic expressions with confidence.

Negative Numbers Division

The division of negative numbers can sometimes confound students, but the rule is quite straightforward: the quotient of two negative numbers is positive. This concept is crucial when simplifying algebraic expressions that involve negative coefficients. The given problem (-4x (-4) is a perfect example of this rule.
In our case, both the numerator and the denominator are -4. According to the division rule for negative numbers, we divide these like we would positive numbers but remember that a negative divided by a negative results in a positive outcome. Therefore, (-4) (-4) simplifies neatly to 1. Remember always to consider the sign of numbers in any arithmetic operation, as it can significantly affect the result. This detail is especially important in the context of algebra, where signs determine the direction of inequalities and the shape of graphs.

Algebraic Expressions Simplification

The simplification of algebraic expressions is a process of reducing complexity while maintaining equivalency. It's a fundamental skill that requires understanding the properties of numbers and operations. In our exercise, the simplification involved is relatively simple: (-4x (-4) simplifies to x since dividing something by itself (except zero) yields one, and multiplying by 1 has no effect on the value.
This step is vital as it makes an expression easier to work with, may unveil further simplification opportunities, and allows us to reach the most 'streamlined' form of an expression. When simplifying, look for common factors, apply the distributive property, and combine like terms when appropriate. Always aim to express your final answer in the most concise form. For example, rather than writing 1x, we simply write x. This principle of expressing in simplest terms is invaluable for clarity and eliminates unnecessary complexity in algebra.