$\frac{1}{y-2}=\frac{2}{y+1}$
Answer (2)
Cross-multiply the equation: 1 ( y + 1 ) = 2 ( y − 2 ) .
Expand both sides: y + 1 = 2 y − 4 .
Isolate y by subtracting y from both sides and adding 4 to both sides: 1 = y − 4 ⟹ y = 5 .
Verify the solution by substituting y = 5 into the original equation, which holds true. The final answer is 5 .
Explanation
Understanding the Problem We are given the equation y − 2 1 = y + 1 2 . Our goal is to find the value of y that satisfies this equation.
Cross-Multiplication To solve for y , we first cross-multiply to eliminate the fractions: 1 × ( y + 1 ) = 2 × ( y − 2 )
Expanding the Equation Next, we expand both sides of the equation: y + 1 = 2 y − 4
Isolating y Now, we want to isolate y on one side of the equation. We can subtract y from both sides: 1 = 2 y − y − 4 1 = y − 4
Solving for y Finally, we add 4 to both sides to solve for y : 1 + 4 = y y = 5
Verification To check our solution, we substitute y = 5 back into the original equation: 5 − 2 1 = 5 + 1 2 3 1 = 6 2 3 1 = 3 1 Since the equation holds true, our solution is correct.
Examples
In real-world scenarios, solving equations like this can help in various fields. For example, in physics, you might use similar equations to determine the position or velocity of an object. In economics, you could use them to find equilibrium points in supply and demand models. Understanding how to manipulate and solve these equations is a fundamental skill in many quantitative disciplines.
To solve y − 2 1 = y + 1 2 , we cross-multiply and simplify to find y = 5 . Verification shows that this solution is correct. Therefore, the final answer is 5 .
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