Problem 42 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ 9(a-4)=0 $$

Step-by-Step Solution

Answer

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Step 1: Write down the given equation

The given equation is $$9(a-4) = 0.$$

Step 2: Apply the zero-factor property

According to the zero-factor property, if a product of two factors is equal to zero, then at least one of the factors must be equal to zero. So, we have two possibilities: 1. Either $$9 = 0$$ or 2. $$(a-4)=0$$

Step 3: Solve for 'a'

Clearly, $$9 \neq 0$$. So, we don't need to consider the first possibility. For the second possibility, we have $$a-4=0$$ Adding 4 to both sides, we get $$a = 4$$ So, the solution to the given equation is $$a=4$$.

Key Concepts

Solving Equations

When you solve an equation, you're finding the value of a variable that makes the equation true. Equations can be simple or complex, but the goal remains the same: isolate the variable. In simpler terms, this means you rearrange and simplify the equation until the unknown variable stands alone on one side of the equation sign, usually '=', with its numerical value on the other side.

For example, in the equation \(9(a-4) = 0\), our task is to find the value of \(a\) that makes the equation true. Here, the equation involves a product that equals zero. This hints towards using a specific property we’ll discuss further. In every step, your goal is to manipulate the equation to eventually solve for the variable. Keep in mind that every operation you perform should maintain the balance of the equation. Whether you're adding, subtracting, multiplying, or dividing, it should be done to both sides. This ensures that the equation stays true and the solution remains valid.

Zero Product Property

The zero product property is a powerful and simple tool in algebra used to solve equations involving products that equal zero. It states that if the product of two or more factors is zero, at least one of the factors must be zero. This is because zero is the only number that, when multiplied by another number, results in zero.

In the problem \(9(a-4)=0\), we apply the zero product property. The idea is:

• Either the first factor, \(9\), equals zero
• or the second factor, \((a-4)\), equals zero

However, since \(9\) is not zero, the only viable option is to set \((a-4)\) equal to zero. This simplifies the equation and leads us to the solution. The utility of the zero product property lies in its ability to break down complex equations into simpler ones, ultimately guiding us to find the values that satisfy the original equation.

Algebraic Equations

Algebraic equations are statements of equality involving algebraic expressions. These equations can have constants, variables, and arithmetic operators like addition, subtraction, multiplication, and division. Solving them involves finding the values of variables that make the equation true.

Take \(9(a-4) = 0\) for instance: it's an algebraic equation involving a variable \(a\). The equation shows a relationship between the variable and constants using multiplication. Solving an algebraic equation typically requires an understanding of various mathematical properties, like the zero product property, as well as operations like adding, subtracting, multiplying, and dividing.

When solving algebraic equations, you use these operations to isolate the variable. In essence, each operation is a small step towards revealing the value of the unknown. The consistency in using these methods is what allows these equations to be solved predictably and accurately, providing precise values for variables in a mathematical sentence.