Problem 35 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ 8(y-6)=0 $$

Step-by-Step Solution

Answer

Question: Solve the equation using the zero-factor property: $$8(y-6) = 0$$ Answer: The equation is true when $$y = 6$$.

Step 1: Identify the factors

The given equation is in the form: $$ 8(y-6)=0 $$ Here, the two factors are: 8 and (y-6).

Step 2: Use the zero-factor property

According to the zero-factor property, if a product of two factors is zero, then at least one of those factors must be zero. So, we set each factor to zero and solve for y. For the first factor: $$ 8 = 0 $$ Since 8 is a constant and not equal to zero, there is no solution for y from this factor. For the second factor: $$ y-6 = 0 $$

Step 3: Solve for y

Add 6 to both sides of the equation to isolate y: $$ y-6+6 = 0+6 $$ Simplifying, we get: $$ y = 6 $$

Step 4: Write the final solution

The solution to the equation is: $$ y = 6 $$ So, the equation is true when y is equal to 6.

Key Concepts

Algebraic Equations

Algebraic equations are expressions that contain variables, constants, and arithmetic operations which are set equal to each other. They represent mathematical relationships and can be linear, quadratic, or higher degree equations. In our exercise, the equation is:
\( 8(y-6) = 0 \)
This is a simple linear algebraic equation where we encounter a variable \( y \) and constants. The goal in solving algebraic equations is to find the value of the variable that satisfies the equation.

• In linear equations, variables like \( y \) appear only to the first power (no squares, cubes, etc.).
• They set two expressions equal to each other, forming a line when graphed on a coordinate plane.

This specific equation highlights a property often used in algebra called the zero-factor property, which simplifies the task of solving by dealing with products that result in zero.

Factorization

Factorization in algebra involves breaking down expressions into products of simpler expressions or factors. This method is essential for solving equations effectively, especially when utilizing the zero-factor property. In our given exercise, the equation:
\( 8(y-6) = 0 \)
is already presented in factored form. The factors are \( 8 \) and \( (y-6) \). Factorization simplifies the problem; instead of dealing with complex expressions, we work with individual factors.

• Identify the factors of an expression to make it easier to solve.
• Factorization helps in revealing solutions when one or more of the factors can be zero.
• For equations set to zero, once in factorized form, one can deduce solutions directly using properties like the zero-factor property.

Understanding how to factorize algebraic expressions is crucial in managing and solving more complex equations.

Solving Linear Equations

Solving linear equations involves finding a value for the variable that makes the equation true. The exercise gives a perfect example of a linear equation in factored form:
\( 8(y-6) = 0 \)
Once in this form, we apply the zero-factor property, which states that if the product of two factors is zero, then at least one factor must be zero.

Steps to Solve

• Identify Factors: The equation has the factors \( 8 \) and \( (y-6) \).
• Apply Zero-Factor Property: Initially, assume each factor might be zero.
• Test Each Factor: \( 8 = 0 \) is false, so no solution here. \( (y-6) = 0 \) leads to solving \( y \).
• Isolate the Variable: Add 6 to both sides of the equation \( y-6 = 0 \) to find \( y = 6 \).

Thus, the solution is \( y = 6 \). Mastering these steps allows one to tackle more challenging equations with confidence and ease.