Problem 10 - Intermediate Algebra

Question

Solve equation. \(\frac{1}{3}-\frac{10}{x}=-3\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = 3 \).

Step 1: Isolate the Variable Term

Start with the equation \( \frac{1}{3} - \frac{10}{x} = -3 \). To isolate the term involving \( x \), subtract \( \frac{1}{3} \) from both sides to get: \( -\frac{10}{x} = -3 - \frac{1}{3} \).

Step 2: Simplify the Right Side

Convert \( -3 \) to a fraction to combine it with \( -\frac{1}{3} \). Since \( -3 = -\frac{9}{3} \), the equation becomes \( -\frac{10}{x} = -\frac{9}{3} - \frac{1}{3} \). Simplify the right side to get: \( -\frac{10}{x} = -\frac{10}{3} \).

Step 3: Solve for x

We have \( -\frac{10}{x} = -\frac{10}{3} \). Multiply both sides by \( x \) to eliminate the fraction: \( -10 = -\frac{10}{3}x \). Then multiply both sides by \( -3 \) to solve for \( x \): \( 30 = 10x \).

Step 4: Finalize the Solution

Divide both sides by 10 to isolate \( x \): \( x = \frac{30}{10} = 3 \).

Key Concepts

Isolate the Variable

When solving rational equations, the first step is often to isolate the variable you are solving for. This involves getting all terms that contain the variable on one side of the equation and all other terms on the opposite side. For the equation \( \frac{1}{3} - \frac{10}{x} = -3 \), notice that the term \( \frac{10}{x} \) includes the variable \( x \). To isolate this term, we move the constant fraction \( \frac{1}{3} \) to the right side by subtracting it from both sides of the equation.

By doing this, you simplify the work you need to do later. Once the terms are moved, the equation transforms into \( -\frac{10}{x} = -3 - \frac{1}{3} \). Isolating the variable is a crucial step because it clarifies the variable's placement, making subsequent steps more straightforward.

Combining Fractions

Combining fractions is another essential skill needed in solving rational equations. When you reach an equation with fractions, it’s important to perform arithmetic with them correctly. In our example, we have \( -3 - \frac{1}{3} \). Because we can rewrite \( -3 \) as \( -\frac{9}{3} \), we can easily combine it with \( -\frac{1}{3} \) since they now have the same denominator.

Make sure when you’re combining fractions to always find a common denominator first. Once the fractions are expressed with the same denominator, simply combine the numerators. In this case, combining gives us \( -\frac{10}{3} \). Turning whole numbers into fractions can simplify the process, as seen here, and is an essential part of solving rational equations.

Simplifying Equations

Simplifying equations helps to make solving easier and more efficient. Once you've isolated the variable term and combined fractions, the next task is to simplify the equation fully. After isolating and combining in our example, the equation becomes \( -\frac{10}{x} = -\frac{10}{3} \).

To proceed, eliminate the fraction by multiplying both sides by \( x \) to clear the variable from the denominator, resulting in \( -10 = -\frac{10}{3}x \). Further simplification comes by multiplying both sides by \(-3\), turning the equation into \( 30 = 10x \). Finally, dividing by 10 gives us \( x = 3 \). This sequence of simplifying ensures the variable stands alone, providing a clear and direct pathway to the solution.