Problem 37 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ -6(n+15)=0 $$

Step-by-Step Solution

Answer

Answer: The solution to the equation is n = -15.

Step 1: Identify the factors of the equation.

In the given equation, we can see that there is a single factor multiplied by -6. $$ -6(n+15) = 0 $$

Step 2: Apply the zero-factor property.

According to the zero-factor property, if the product of factors is equal to 0, then at least one of the factors has to be 0. The factors here are -6 and (n+15). The equation can be written as: $$ -6 \cdot (n + 15) = 0 $$ To satisfy the zero-factor property, either -6 can be equal to 0 or (n+15) can be equal to 0.

Step 3: Set each factor equal to 0 and solve for n.

-6 can never equal 0. So, (n+15) must be equal to 0. $$ n + 15 = 0 $$ Now, we need to solve for n. $$ n = -15 $$

Step 4: Write the solution.

The solution to the equation is n = -15.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra used to find the value of an unknown variable. In simple terms, it involves working with equalities where different expressions are set equal to each other. The main goal is to isolate the unknown variable, typically denoted by letters like \( n \), \( x \), or \( y \). This process involves several techniques, like:

• Balancing both sides of the equation.
• Applying algebraic operations such as addition, subtraction, multiplication, or division.
• Using properties like the zero-factor property to simplify the problem.

Breaking down a problem into smaller, manageable steps is crucial for solving equations efficiently. With practice, this helps in understanding the relationship between different parts of the equation and ultimately finding the correct variable value.

Elementary Algebra

Elementary algebra lays the groundwork for all higher-level mathematics. It is centered around the use of algebraic symbols and provides methods for manipulating expressions and solving equations. Key concepts include:

• Variables: Symbols that stand in for unknown values.
• Constants: Fixed numerical values like \( -6 \) or \( 15 \) as seen in the original exercise.
• Coefficients: Numbers that multiply variables, such as \( -6 \) in \(-6(n+15)\).
• Expressions: Combinations of variables, constants, and coefficients.

In elementary algebra, understanding these components helps in building the ability to form, simplify, and solve equations. This forms the essential skill set needed for tackling increasingly complex algebraic problems as a student progresses in mathematics.

Factoring

Factoring is a method used in algebra to simplify expressions or solve equations. The idea is to write a number or expression as a product of its factors. In the zero-factor property, if a product of factors equals zero, at least one of the factors must also equal zero. Factoring helps break down expressions into more manageable pieces, making it easier to solve problems.

The Zero-Factor Property

The zero-factor property is particularly crucial when dealing with quadratic equations or expressions like \(-6(n+15) = 0\). According to this property:

• If \( ab = 0 \), then at least one of \( a \) or \( b \) must be zero.
• This allows you to split the equation and solve for each factor separately.

By setting each factor equal to zero, you can solve for the unknown variable efficiently as shown in the solution provided. Being comfortable with factoring is an essential tool for students to understand and solve algebraic equations effectively.