Solve [tex]\frac{k+1}{2 k-1}=\frac{4}{5}[/tex].
Answer (1)
To solve the equation:
Cross-multiply: 5 ( k + 1 ) = 4 ( 2 k − 1 ) .
Expand: 5 k + 5 = 8 k − 4 .
Rearrange: 3 k = 9 .
Divide: k = 3 .
The solution is 3 .
Explanation
Understanding the Problem We are given the equation 2 k − 1 k + 1 = 5 4 . Our goal is to solve for k .
Cross-Multiplication To solve the equation, we can cross-multiply. This means we multiply both sides of the equation by 5 ( 2 k − 1 ) to eliminate the fractions: 5 ( k + 1 ) = 4 ( 2 k − 1 ) .
Expanding Both Sides Next, we distribute the numbers on both sides of the equation: 5 k + 5 = 8 k − 4 .
Rearranging the Equation Now, we want to isolate the k terms on one side and the constant terms on the other side. We can subtract 5 k from both sides and add 4 to both sides: 5 k + 5 − 5 k + 4 = 8 k − 4 − 5 k + 4 which simplifies to 9 = 3 k .
Solving for k To solve for k , we divide both sides by 3: 3 9 = 3 3 k which simplifies to k = 3 .
Final Answer Therefore, the solution to the equation is k = 3 .
Examples
Imagine you're scaling a recipe. The original recipe makes a certain amount, but you need to adjust the quantities to make a different amount. Solving equations like this helps you determine the new quantities of ingredients needed to maintain the correct ratios and proportions in your adjusted recipe. This ensures that your scaled recipe turns out just as delicious as the original!
