Problem 99 - Algebra for College Students

Question

Your classmate solves the equation \(3 a x+b x=0\) for \(x\) as follows: $$ \begin{aligned} 3 a x+b x &=0 \\ 3 a x &=-b x \\ x &=\frac{-b x}{3 a} \end{aligned} $$ How should he know that the solution is incorrect? How would you help him obtain the correct solution?

Step-by-Step Solution

Answer

Simplify to \((3a+b)x = 0\), and \(x = 0\) is a valid solution.

Step 1: Check the Initial Equation Setup

The original equation given is \(3ax + bx = 0\). This is the correct starting point for solving the equation.

Step 2: Simplify Equation by Combining Like Terms

Combine the terms involving \(x\) on the left-hand side: \((3a + b)x = 0\). Here, we factor \(x\) out of both terms.

Step 3: Use the Zero Product Property

When a product of two factors equals zero, one or both of the factors must be zero. So, set each factor equal to zero separately. Thus, \((3a + b) = 0\) or \(x = 0\).

Step 4: Solve for Unknowns

Solve for \(x\) which gives \(x = 0\). Also, solve \(3a + b = 0\) for \(b\) in terms of \(a\) if needed: \(b = -3a\).

Step 5: Identify Mistakes in Original Solution

The mistake in the original solution is incorrectly rearranging the terms by moving \(bx\) to the right-hand side, which does not solve the equation correctly. Additionally, dividing by \(x\) prematurely while \(x\) could be zero is also a mistake.

Step 6: Ensure Proper Solution Understanding

The key process is to combine like terms first and apply the zero product property correctly, which should be emphasized for solving similar equations.

Key Concepts

Zero Product Property

The zero product property is a fundamental principle used in solving linear equations that involve factoring. It states that if a product of two numbers is zero, then at least one of the numbers must be zero. For example, if we have an equation like

• \((a\cdot b = 0)\),

it must be that either

• \(a = 0\) or
• \(b = 0\).

This concept is crucial because it allows us to set each factor in the equation individually to zero and solve for unknowns. In the given exercise of solving \((3ax + bx = 0)\), the zero product property lets us write

• \((3a + b) = 0\) or
• \(x = 0\).

This helps in breaking down the equation into simpler, more manageable parts.

Factoring

Factoring is the process of breaking down an expression into a product of its simplest components, which are called factors. In the exercise provided, we see the equation as \((3ax + bx)\).
Recognizing that both

• terms contain a common factor \(x\),

we factor \(x\) out from each term to make the equation easier to solve:

• \((3a + b)x = 0\).

Factoring does not solve the equation directly but helps simplify it to use the zero product property. Properly applying factoring can transform an equation into a form where solutions are more straightforward to find.
Even though it seems simple, factoring out terms correctly is crucial to avoid errors in subsequent steps.

Combining Like Terms

When solving equations, combining like terms means merging terms that have the same variable component and can be operated on together. These are terms with identical variables raised to the same power. For example, in the equation \((3ax + bx = 0)\),we observe that both terms have

• \(x\) as a factor.

Thus, we can combine them as

• \((3a + b)x\).

Combining like terms simplifies the equation, making it easier to work with. This step is essential as it helps organize and simplify the equation, ensuring it is in the optimal form for solving using other algebraic techniques like factoring or the zero product property.
A clear understanding of this step prevents common mistakes in moving terms around or misapplying arithmetic rules.

Equation Simplification

Equation simplification involves performing operations to make the equation easier to solve. This can include actions like combining like terms, factoring, and using properties such as the zero product property. In the original mistake, a step presumed we could divide by terms involving the variable \(x\),

• \(x = \frac{-b x}{3 a}\),

before correctly organizing the equation was an incorrect simplification.
Instead, by correctly combining terms to form

• \((3a + b)x = 0\)

we simplify using zero-product property. Simplification is a tool to bring clarity and structure to an equation, ensuring it is presented in the simplest form so that solving becomes a logical and methodical process.