Problem 32 - Elementary Algebra

Question

Solve each of the equations. $$\frac{-3}{2 x-5}=\frac{-4}{x-3}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{11}{5}\).

Step 1: Set Up the Equation

The given equation is \( \frac{-3}{2x-5} = \frac{-4}{x-3} \). Both sides are fractions, so we can solve this equation by cross-multiplying to eliminate the fractions.

Step 2: Cross-Multiply

Cross-multiply to get: \[-3(x-3) = -4(2x-5)\]

Step 3: Simplify Both Sides

Distribute the terms on each side of the equation:- Left Side: \(-3 \cdot x + 9 = -3x + 9\)- Right Side: \(-4 \cdot 2x + 20 = -8x + 20\)

Step 4: Equate and Simplify

Now, set the simplified expressions equal to each other:\(-3x + 9 = -8x + 20\)

Step 5: Solve for x

To isolate \(x\), first add \(8x\) to both sides:\(-3x + 8x + 9 = 20\)which simplifies to:\[5x + 9 = 20\]Subtract 9 from both sides:\[5x = 11\]Finally, divide both sides by 5 to solve for \(x\):\[x = \frac{11}{5}\]

Step 6: Verify the Solution

Substitute \(x = \frac{11}{5}\) back into the original equation to verify it satisfies both sides of the equation. Check if: \( \frac{-3}{2 \cdot \frac{11}{5} - 5} = \frac{-4}{\frac{11}{5} - 3} \). Simplifying the fractions confirms the equation holds true, verifying \(x = \frac{11}{5}\) is correct.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions. It allows us to eliminate the fractions by multiplying across the equation, hence the name "cross-multiplication." This method works because when two fractions are equal, their cross-products are also equal. Consider the equation \( \frac{a}{b} = \frac{c}{d} \).

To find the value of the unknown, you multiply the numerator of one fraction by the denominator of the other. So, you multiply \( a \times d \) and \( b \times c \), and set these products equal:

• \( a \times d = b \times c \)

By doing this, you form a simpler equation without fractions, making it much easier to solve. This process reduces the two fractions to a linear equation in most cases, which is straightforward to handle.

Equation Solving

Equation solving involves finding the value of the unknown variable that makes the equation true. It requires careful manipulation of the equation to isolate the variable on one side. When you have a linear equation like \(-3x + 9 = -8x + 20\), your goal is to solve for \(x\).To achieve this, follow a few logical steps:

• First, eliminate any constants or numbers from the side of the equation where your variable is. You can do this by adding or subtracting them from both sides.
• Next, if your unknown has a coefficient, divide both sides of the equation by this coefficient to solve for the variable.
• It is essential to ensure that the operations you perform are balanced on both sides to maintain equivalence.

By following these steps, you systematically find the solution to the equation, ensuring that it is correct through validation if necessary.

Fractions in Equations

Fractions in equations can appear daunting at first, but they can be tackled confidently by using strategies like cross-multiplication.Fractions occur when part of an expression is divided by another, such as \( \frac{a}{b} \).

Here are some tips for handling fractions in algebraic equations:

• Look for a common denominator if multiple fractions are present, which can simplify combining them.
• Consider multiplying all terms by the least common denominator to eliminate fractions altogether.
• Using cross-multiplication, as described earlier, is a valuable method for proportion-based equations.

By understanding how to manipulate and simplify fractions, you can solve equations more efficiently, improving your overall algebra skills.