Problem 24 - Elementary Algebra

Question

Solve. $$ 5(7 x-8) 2=0 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{8}{7} \).

Step 1: Simplify the Equation

We start with the equation \( 5(7x - 8)2 = 0 \). Combine the constants outside the parenthesis: \( 5 \times 2 = 10 \). So, the equation becomes \( 10(7x - 8) = 0 \).

Step 2: Apply the Zero Product Property

According to the zero product property, if a product of two factors is zero, then at least one of the factors must be zero. Hence, we set the expression inside the parenthesis to zero: \( 7x - 8 = 0 \).

Step 3: Solve for x

To isolate \( x \), we add 8 to both sides of the equation: \( 7x = 8 \). Then, divide both sides by 7 to solve for \( x \): \( x = \frac{8}{7} \).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that simplifies the process of solving equations involving products. This property states that if the product of two or more factors is zero, at least one of those factors must be zero. This is a key concept when dealing with equations in the form of a product equal to zero, like in our exercise with the equation \( 10(7x - 8) = 0 \).
To apply this property:

• Focus on each factor of the product. In this example, the product involves the terms \( 10 \) and \( (7x - 8) \).
• Since the constant \( 10 \) cannot be zero, the only factor that can lead to the product being zero is the expression \( (7x - 8) \).
• According to the zero product property, set the non-zero factor equal to zero: \( 7x - 8 = 0 \).

Understanding this property allows us to break down complex equations into simpler parts, making it easier to isolate and solve for the variables involved.

Simplifying Equations

Simplifying equations is an essential step in solving any algebraic equation. It involves reducing the equation to its most basic form. This helps us clearly see the relationship between the variables and constants.
In the given exercise, simplifying begins by identifying and combining like terms and constants. The original equation has a multiplication outside the parenthesis: \( 5(7x - 8)2 \). To simplify:

• Multiply the constants \( 5 \) and \( 2 \) to get \( 10 \).
• Replace the original expression in the equation with the simplified form \( 10(7x - 8) \).

This step effectively reduces the equation to an easier-to-handle expression. Simplifying is crucial as it often leads to quicker and more accurate solutions, avoiding unnecessary complexity or mistakes in calculations.

Isolating Variables

Isolating variables is the process of manipulating an equation so that the variable we are solving for stands alone on one side of the equation. It is a critical skill allowing us to find the value of unknowns.
In our example, the goal is to solve for \( x \) in the equation \( 7x - 8 = 0 \). To isolate \( x \), follow these steps:

• Add 8 to both sides of the equation to cancel out the subtraction: \( 7x = 8 \).
• Next, divide both sides by 7 to solve for \( x \): \( x = \frac{8}{7} \).

By systematically undoing the operations affecting \( x \), we isolate it effectively. This step-by-step process helps ensure that the final solution is correct and reflects the mathematics accurately. Through isolating, you address each operation on a variable logically, always aiming to simplify its surroundings until the variable stands alone.