Problem 64 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$\frac{3 x}{4.5}=\frac{1}{8}$$

Step-by-Step Solution

Answer

After completing the steps, the value of \(x\), rounded to two decimal places is 0.19.

Step 1: Clear the Fraction

Start by multiplying every term by 4.5 to eliminate the denominator on the left side. This results in the equation \(3x = \frac{1}{8} * 4.5\).

Step 2: Solve for the Variable

Next, divide each side by 3 to solve for \(x\). This gives \(x = \frac{1}{8} * 4.5 / 3\).

Step 3: Simplify the Expression

Simplify the expression on the right and round to two decimal places to get the final value for \(x\).

Key Concepts

Algebraic Fractions

Algebraic fractions, simply put, are fractions that contain a variable, like the one in the exercise \( \frac{3x}{4.5} = \frac{1}{8} \). Understanding how to work with these fractions is critical for solving algebraic equations. Students often need to manipulate algebraic fractions to eliminate the denominators so that the equation only contains numerators and, subsequently, easier to solve. A common approach is to find a common denominator, or in cases with simpler equations, to multiply both sides of the equation by the denominator to clear it completely. This method transforms an algebraic fraction into a simpler linear equation, paving the way for easier variable isolation and solution.

Equation Solving Steps

The process of solving equations, especially those involving algebraic fractions, involves several key steps. Here's a breakdown of a general approach:

• Clearing Fractions: Multiply every term by the least common denominator to remove the fraction, as done in the exercise by multiplying with 4.5.
• Isolating the Variable: Rearrange the equation to get the variable on one side. This simplifies the equation to a form where the variable stands alone.
• Simplification: Carry out arithmetic operations to simplify the expression.

Solving linear equations might seem daunting at first, especially with fractions involved, but following these structured steps can demystify the process.

Decimal Approximation

In some cases, like with our previous exercise, the final answer isn't a neat whole number but a decimal. Approximating the answer to a certain number of decimal places may be necessary. For instance, the exact answer to \( \frac{1}{8} * 4.5 / 3 \) might not be easily interpretable or usable, especially in real-world applications; hence, rounding it off to two decimal places makes it more practical. This rounding off is what we refer to as decimal approximation and is particularly useful for providing succinct, readable answers in calculations that involve measurements or finances.

Variable Isolation

Variable isolation is the paramount goal when solving equations - it is the step where you ultimately solve for the unknown. This involves manipulating the equation such that the variable is by itself on one side of the equal sign. For the provided equation, dividing by the coefficient 3 isolates the variable \( x \) on one side, resulting in \( x = \frac{1}{8} * 4.5 / 3 \). The essence of this step is to unravel the value of the variable without any other numeric or algebraic terms attached to it. Mastery of this technique allows students to unlock the value of unknowns in a wide array of mathematical problems.