Problem 29 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{4}{5 y-1}=\frac{2}{2 y-1}$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = -1 \).

Step 1: Cross-Multiply

Cross-multiply the fractions to remove the denominator: \( 4*(2y-1) = 2*(5y-1) \). This simplifies to \( 8y-4 = 10y-2 \).

Step 2: Simplify the Equation

Rearrange terms to one side to simplify equation. This involves subtracting \( 8y \) from both sides to get: \(0 = 2y+2 \).

Step 3: Solve for y

Finally, solve for \( y \) by subtracting 2 from both sides and dividing through by 2: \((0-2)/2 = y\), meaning \( y = -1 \).

Key Concepts

Cross-multiplication

Cross-multiplication is a valuable technique used to simplify and solve rational equations. It eliminates the fractions by multiplying each numerator by the denominator of the opposite fraction.
This method is especially useful when dealing with two fractions set equal to each other.

• Consider the equation \( \frac{a}{b} = \frac{c}{d} \).
• Cross-multiplication results in \( a \times d = b \times c \).
• This process turns the equation into a simpler form without fractions, making it easier to solve.

In our exercise, the equation \( \frac{4}{5y-1} = \frac{2}{2y-1} \) becomes \( 4 \times (2y-1) = 2 \times (5y-1) \) after cross-multiplying.
This leads us to the equation \( 8y - 4 = 10y - 2 \), which can be further simplified to find the value of \( y \).

Equation solving

Solving equations involves finding the values of variables that satisfy the equation. Once cross-multiplication has simplified the equation, the next step is solving it.
Here are some useful tips:

• Start by gathering all variable terms to one side of the equation, and constant numbers to the other.
• Use basic arithmetic operations: addition, subtraction, multiplication, and division.
• In our example, subtract \( 8y \) from both sides to collect \( y \) terms together. This brings the equation to \( 0 = 2y + 2 \).
• Next, isolate \( y \) by subtracting 2 from both sides, giving \( -2 = 2y \).
• Finally, divide by 2 to solve for \( y \), resulting in \( y = -1 \).

Each step moves you closer to solving for the variable, providing a clear path to finding the solution.

Rational equations

Rational equations involve fractions that contain variables in the denominators. Solving these equations requires steps to clear fractions and then solve for the variables like any other equation.
They often appear intimidating, but with clear steps, they become manageable.

• Begin by using cross-multiplication to remove fractions, as shown in the first step.
• Ensure to consider any restrictions on the variable to keep denominators non-zero.
• In our exercise, both \(5y-1\) and \(2y-1\) must not equal zero, thus \(y\) cannot be \(\frac{1}{5}\) or \(\frac{1}{2}\).
• Once fractions are removed, treat them like linear equations and solve as previously described.

Keep practicing rational equations to improve your ease and efficiency with these types of problems. Understanding and applying these techniques will make solving them less daunting.