Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(0.04(x-2)=0.02(6 x-3)-0.02\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.25\).

Step 1: Distribute the Parentheses

Multiply the 0.04 with the \(x-2\), and 0.02 with \(6x-3\). This produces the following equation: \(0.04x - 0.08 = 0.12x - 0.06 - 0.02\).

Step 2: Combine Like-Terms

Combine the like terms on each side of the equality. To achieve this, subtract \(0.04x\) from both sides of the equation while adding 0.08 to both sides. We get \(0.08x = 0.02\).

Step 3: Isolate the Variable x

To get the variable by itself, divide both sides of the equation by the coefficient of \(x\), 0.08: \(x = \frac{0.02}{0.08} = 0.25\).

Key Concepts

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. When faced with an equation, our goal is to isolate the variable, typically denoted as \( x \), to one side of the equation. This way, we can find its value.
It's like a puzzle; we're trying to find out what number \( x \) represents to keep both sides equal. When solving any equation, the result could lead to:

• A specific solution, where you find an exact value for \( x \).
• No solution, which happens if no possible number makes the equation true.
• An infinite number of solutions, meaning any real number will satisfy the equation.

Approaching the problem systematically by applying algebraic principles can always help us come to the correct conclusion.

Distributive Property

The distributive property is a key algebraic principle used to eliminate parentheses in an equation. It states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) equals \( ab + ac \).
In our original exercise, we used the distributive property to handle expressions like \( 0.04(x-2) \) and \( 0.02(6x-3) \). Here's how it was applied:

• The expression \( 0.04(x-2) \) becomes \( 0.04x - 0.08 \).
• The expression \( 0.02(6x-3) \) turns into \( 0.12x - 0.06 \).

By distributing the number outside the parentheses across each term within the parentheses, we simplify the equation. This step is crucial because it paves the way for combining like terms and further simplification.

Combining Like Terms

After employing the distributive property, we often face an equation with multiple similar terms, such as multiple \( x \) terms or constant terms on one or both sides of the equation. Combining like terms means you group these similar terms together.
In our equation, we started with \( 0.04x - 0.08 = 0.12x - 0.06 - 0.02 \). To simplify:

• Subtract \( 0.04x \) from both sides to move all \( x \) terms to one side.
• Add 0.08 to both sides to consolidate constant terms.

This leads to the equation \( 0.08x = 0.02 \). By combining like terms, the equation becomes simpler and more manageable, making the path to solving for \( x \) clearer.

Isolating Variables

The ultimate goal of solving an equation is usually to isolate the variable. Isolating the variable involves performing algebraic operations to get the variable alone on one side of the equation. This step allows us to directly determine its value.
In the exercise, we have the equation \( 0.08x = 0.02 \). To isolate \( x \), we need to remove the coefficient 0.08. We do this by dividing both sides of the equation by 0.08.

• This division gives us \( x = \frac{0.02}{0.08} \).
• Finally, we simplify this fraction to find \( x = 0.25 \).

By isolating the variable, we've successfully solved the equation and found the value of \( x \). This process requires careful manipulation to maintain the equation's balance. This approach applies to any equation where we're looking for a variable's value.