Problem 19 - Elementary Algebra

Question

Solve each of the equations. $$\frac{3}{2 x-1}=\frac{2}{3 x+2}$$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{8}{5} \).

Step 1: Set Up the Equation

We need to solve the equation \( \frac{3}{2x-1} = \frac{2}{3x+2} \). The goal is to find the value of \( x \) that makes this equation true.

Step 2: Cross-Multiply

To eliminate the fractions, cross-multiply the terms. Multiply each numerator by the opposite denominator:\[ 3(3x + 2) = 2(2x - 1) \]

Step 3: Distribute

Distribute the constants across the terms:\[ 3 \times 3x + 3 \times 2 = 2 \times 2x - 2 \times 1 \]This simplifies to:\[ 9x + 6 = 4x - 2 \]

Step 4: Isolate Variables

Get all the terms with \( x \) on one side of the equation. Subtract \( 4x \) from both sides:\[ 9x - 4x + 6 = -2 \]This simplifies to:\[ 5x + 6 = -2 \]

Step 5: Solve for x

Subtract 6 from both sides to solve for \( x \):\[ 5x = -2 - 6 \]This simplifies to:\[ 5x = -8 \]

Step 6: Find the Value of x

Divide both sides by 5 to isolate \( x \):\[ x = -\frac{8}{5} \]

Step 7: Verify the Solution

To ensure the solution is correct, substitute \( x = -\frac{8}{5} \) back into the original equation and verify both sides are equal. If they are, the solution is verified. In this case, the substitution confirms validity, as both sides reduce to equal values.

Key Concepts

Cross-Multiplication

Cross-multiplication is a vital technique when solving rational equations that involve fractions. It allows us to eliminate the denominators, making equations easier to handle. In equations like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other. This results in the equation \( a \times d = b \times c \).

This step is crucial since it transforms the given fraction problem into a simple algebraic equation, thus, easing the solution process. Always ensure each term of the original fractions is correctly multiplied to avoid any potential errors. With practice, this process becomes intuitive, laying a strong foundation for tackling more complex problems.

Distributing Terms

After employing cross-multiplication, the next step is distributing terms. Distribution entails expanding expressions by multiplying each term in a bracket by the factor outside the bracket. This is essential to simplify the expressions by removing parentheses.

For instance, in our example, once cross-multiplication is applied, we have the expression \( 3(3x + 2) = 2(2x - 1) \). Distributing involves multiplying 3 by each term inside the first bracket, yielding \( 9x + 6 \), and similarly, multiplying 2 by each term inside the second bracket, giving \( 4x - 2 \).

By distributing the terms correctly, you ensure the equation is simplified accurately, which is a crucial step before proceeding to solve the equation.

Isolating Variables

Once the equation is simplified, the next task is isolating the variable. Isolating the variable involves rearranging the equation to have the unknown on one side and the constants on the other. This is often done by using inverse operations.

In our specific problem, after distributing terms, the resulting equation is \( 9x + 6 = 4x - 2 \). To isolate \( x \), subtract \( 4x \) from both sides, leading to \( 5x + 6 = -2 \). The next step is to move the constant to the other side by subtracting 6 from both sides, resulting in \( 5x = -8 \).

This process is critical as it prepares the equation for the final balance, gearing up for the precise calculation of the variable's value. Ensure to perform the same operation on both sides to maintain the equality.

Verifying Solutions

Finally, verifying solutions is a pivotal step, ensuring the derived solution satisfies the original equation. This involves substituting the found value back into the initial equation and checking whether both sides are equal.

For our solution \( x = -\frac{8}{5} \), substitute back into the original equation \( \frac{3}{2x-1} = \frac{2}{3x+2} \). After substitution and simplification, verify if both expressions yield the same value. In our case, they do, confirming the solution is correct.

Verification is essential, not only to prevent errors but also to confirm confidence in the problem-solving process. Never skip this step, as it secures the accuracy and reliability of your solution.