Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{3}{7 x-2}=\frac{9}{3 x+1}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{7}{18} \).

Step 1: Cross-Multiply

To solve the equation \( \frac{3}{7x-2} = \frac{9}{3x+1} \), we start by cross-multiplying to eliminate the fractions. This results in \( 3(3x+1) = 9(7x-2) \).

Step 2: Distribute Terms

Next, distribute the constants across the parentheses: \( 3 \times 3x + 3 \times 1 = 9 \times 7x - 9 \times 2 \). This simplifies to \( 9x + 3 = 63x - 18 \).

Step 3: Combine Like Terms

We proceed by moving all terms involving \( x \) to one side and constant terms to the other. Subtract \( 9x \) from both sides: \( 3 = 54x - 18 \).

Step 4: Isolate the Variable

Add 18 to both sides to isolate the term with \( x \): \( 3 + 18 = 54x \) which simplifies to \( 21 = 54x \).

Step 5: Solve for x

Finally, solve for \( x \) by dividing both sides of the equation by 54: \( x = \frac{21}{54} \). Simplifying this fraction, we get \( x = \frac{7}{18} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a helpful technique used to solve equations that involve fractions. When you encounter an equation where each side is a fraction like \( \frac{3}{7x-2} = \frac{9}{3x+1} \), cross-multiplication helps eliminate the fractions, making the equation easier to handle.
To cross-multiply, you multiply the numerator of one fraction by the denominator of the other. In this case:

• Multiply the numerator 3 by the denominator \(3x+1\)
• Multiply the numerator 9 by the denominator \(7x-2\)

This results in the equation \(3(3x+1) = 9(7x-2)\).

This is beneficial because it turns the equation into a more straightforward algebraic expression without fractions, making it easier to manipulate and solve further.

Distributive Property

The distributive property is a handy algebraic rule that states that a term can be distributed across terms inside a parenthesis for multiplication. In our case, after cross-multiplying, we have the equation \( 3(3x+1) = 9(7x-2) \).

Using the distributive property, we distribute the constants (3 and 9) across the parentheses:

• \( 3 \times 3x + 3 \times 1 = 9 \times 7x - 9 \times 2 \)

This simplifies to \( 9x + 3 = 63x - 18 \).

Applying the distributive property ensures each term inside the parentheses is multiplied correctly, which is a crucial foundation for simplifying equations and solving them accurately.

Combining Like Terms

In algebra, it's essential to combine like terms to consolidate and simplify equations. This involves moving and combining terms with the same variable on one side of the equation, and constant terms on the other side.
By rearranging \( 9x + 3 = 63x - 18 \), we aim to get all the \( x \) terms on one side and numeric values on the other. Subtracting \( 9x \) from both sides gives us \( 3 = 54x - 18 \).

Once similar terms are grouped, equations become much easier to solve. Combining like terms is a fundamental step in simplifying complex algebraic expressions.

Solving for Variables

Solving for variables is the process of isolating the variable to find its value. After simplifying the equation to \( 3 = 54x - 18 \), we further isolate \( x \) by getting rid of other numbers on the same side.
Add 18 to both sides to move the constant term: \( 3 + 18 = 54x \). This simplifies to \( 21 = 54x \).
Finally, to get \( x \) by itself, divide both sides by 54, resulting in \( x = \frac{21}{54} \). By simplifying the fraction, we find \( x = \frac{7}{18} \).

This final step of solving for \( x \) lets you identify the value that satisfies the original equation, using skills to manipulate and simplify the equation's structure.