Problem 13 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{6}{x+3}=\frac{4}{x-3}$$

Step-by-Step Solution

Answer

The solution to the rational equation \(\frac{6}{x+3}=\frac{4}{x-3}\) is \(x = 15\).

Step 1: Cross-multiply

Cross-multiply the two fractions to get rid of the denominator. This is done by multiplying the numerator of the first fraction by the denominator of the second, and the denominator of the first by the numerator of the second, which results in the equation: \(6 * (x - 3) = 4 * (x + 3)\).

Step 2: Distribute

Now, distribute the numbers through the parentheses: \(6x - 18 = 4x + 12\).

Step 3: Move terms

To isolate the variable \(x\), move all \(x\) terms to one side of the equation and the constants to the other. This results in the equation: \(6x - 4x = 12 + 18\). Then simplify to: \(2x = 30\).

Step 4: Solve

Finally, solve for \(x\) by dividing both sides of the equation by 2. This gives \(x = 30 / 2 = 15\).

Key Concepts

Cross-Multiplication

When solving rational equations, cross-multiplication is a helpful technique to eliminate the fractions and simplify the problem. It's a method where you multiply diagonally across the equals sign, pairing the numerator of one fraction with the denominator of the other. For instance, with the equation
\[\frac{6}{x+3}=\frac{4}{x-3}\]
you would multiply 6 by \((x - 3)\) and 4 by \((x + 3)\), effectively setting up an equation without fractions:
\[6 \times (x - 3) = 4 \times (x + 3)\]
This is often the first step in dealing with equations that involve fractions because it simplifies the overall structure, turning it into a more manageable linear equation that can be solved through further algebraic manipulations.

Distributive Property

The distributive property allows you to multiply a single term across terms inside a parenthesis. For example, taking the result of cross-multiplication from our problem,
\[6 \times (x - 3) = 4 \times (x + 3)\]
we apply the distributive property to both sides of the equation. Multiplying 6 by both \(x\) and \(-3\), and likewise with 4 by \(x\) and \(3\), we get:
\[6x - 18 = 4x + 12\]
By distributing the coefficients, we've now removed the parentheses and are left with an equation that is easier to solve. The distributive property is an indispensable tool in algebra that helps transform complex problems into simpler ones by spreading a multiplier over multiple addends.

Isolating Variables

To find the value of the unknown in an equation, we need to isolate the variable. This involves rearranging the equation to have the variable we're solving for on one side, and everything else on the other. In the context of our problem,
\[6x - 18 = 4x + 12\]
we want to isolate \(x\). This means getting all the \(x\)-terms on one side and constants on the other. This is done by performing inverse operations, such as adding or subtracting terms from both sides to keep the equation balanced. We subtract \(4x\) from both sides and add \(18\) to both, leading to:
\[2x = 30\]
Now that the variable is isolated, it's straightforward to solve for \(x\) by dividing both sides by 2, giving us:
\[x = 15\]
Isolating the variable is the core strategy for solving most algebraic equations; it turns a complex problem into a simple matter of arithmetic.