How many solutions does this equation have?
[tex]2 n+5(-2 n-5)=18-8 n+7[/tex]
Answer (2)
Expand the left side of the equation: 2 n + 5 ( − 2 n − 5 ) = − 8 n − 25 .
Simplify the right side of the equation: 18 − 8 n + 7 = 25 − 8 n .
Combine the results: − 8 n − 25 = 25 − 8 n .
Simplify the equation to − 25 = 25 , which is a false statement, indicating that there is no solution. Therefore, the equation has no solution .
Explanation
Analyzing the Equation First, let's analyze the given equation: 2 n + 5 ( − 2 n − 5 ) = 18 − 8 n + 7 . We need to determine how many solutions this equation has. The options are no solution, one solution, or infinitely many solutions.
Expanding the Left Side Next, we will expand the left side of the equation: 2 n + 5 ( − 2 n − 5 ) = 2 n − 10 n − 25 = − 8 n − 25
Simplifying the Right Side Now, let's simplify the right side of the equation: 18 − 8 n + 7 = 25 − 8 n
Combining the Results So, the equation becomes: − 8 n − 25 = 25 − 8 n
Isolating Constants Now, let's add 8 n to both sides of the equation: − 8 n − 25 + 8 n = 25 − 8 n + 8 n − 25 = 25
Determining the Number of Solutions Since − 25 = 25 is a false statement, there are no values of n that can make the original equation true. Therefore, the equation has no solution.
Final Answer Therefore, the equation 2 n + 5 ( − 2 n − 5 ) = 18 − 8 n + 7 has no solution.
Examples
Consider a situation where you are trying to balance a budget. If the equation represents the balance between income and expenses, and you find that the equation has no solution, it means that no matter how you adjust your spending (represented by the variable 'n'), you will never be able to balance your budget. This understanding helps you realize that you need to make fundamental changes to either your income or expenses to achieve financial stability. This type of problem appears in various fields like economics, engineering, and computer science, where balancing equations is crucial for system stability and optimization.
The equation 2 n + 5 ( − 2 n − 5 ) = 18 − 8 n + 7 simplifies to a false statement, − 25 = 25 , indicating it has no solutions. Therefore, the answer is no solution .
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