Problem 39 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (2 x-4)(-3) $$

Step-by-Step Solution

Answer

\(-6x + 12\)

Step 1: Identification

Identify the terms inside and outside the parentheses, namely \(2x-4\) inside and \(-3\) outside.

Step 2: Apply the Distributive Property

Distribute \(-3\) across each term inside the parentheses, which means multiplying \(-3\) with each term inside the parentheses. We get \(-3*2x + -3*-4\).

Step 3: Simplify the Expression

Upon simplifying, \(-3*2x\) is \(-6x\) and \(-3*-4\) is \(12\). So, this brings the expression to \(-6x + 12\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent mathematical relationships. Variables are symbols like \(x\) or \(y\) that can take on various values. These variables are often paired with numbers, called coefficients, and connected through operations such as addition, subtraction, multiplication, and division.

For example, in the expression \(2x - 4\), \(2\) is the coefficient of \(x\), and \(-4\) is a constant term. This expression tells us that \(x\) is being multiplied by 2, and then 4 is subtracted from the product. Algebraic expressions can be evaluated by substituting specific values for the variables or manipulated through algebraic rules to simplify or solve equations.

• Variables can change based on the context or the problem requirements.
• Coefficients are the numbers multiplying the variables.
• Constant terms are numbers in the expression without accompanying variables.

Simplifying Expressions

Simplifying expressions involves reducing them to a form that is easier to work with, often by combining like terms or applying mathematical properties. In our example, we apply the distributive property to remove parentheses and rewrite the expression in a simpler form. This process helps in solving equations and making sense of complex algebraic expressions.

When simplifying, it’s crucial to perform operations correctly. In the expression \((-3)*(2x - 4)\), we need to distribute \(-3\) to each term within the parentheses:

• Multiply \(-3\) by \(2x\) to get \(-6x\).
• Multiply \(-3\) by \(-4\) to get \(+12\).

By completing these steps accurately, the expression simplifies from \((-3)*(2x - 4)\) to \(-6x + 12\).

Remember, simplification not only makes expressions neater but also prepares them for further algebraic operations, such as solving equations or substituting variables later on.

Multiplication of Terms

Multiplication of terms is a fundamental algebraic operation where numbers and variables are multiplied together in different ways. Terms, in this context, refer to parts of an expression separated by addition or subtraction. In the initial stage of our problem, the multiplication involved the term outside the parentheses with each term inside the parentheses.

This multiplication requires following precise rules:

• Multiply the coefficients directly, e.g., \(-3 * 2\) gives \(-6\).
• If variables are involved, their exponents are added, but in our case, since we only have \(x\), the variable remains \(x\).
• Signs matter: multiplying a negative by a positive results in a negative; a negative by a negative results in a positive, as shown in \(-3 * -4 = 12\).

Understanding these rules ensures accurate expansion of expressions and is essential for effectively applying the distributive property, ultimately simplifying expressions or solving equations.