What describes the solution to -10n + 3(8 + 8n) = -6(n - 4)?
Understand the Problem
The question is asking for the nature of the solutions to a given mathematical equation. It requires an understanding of how to manipulate and solve algebraic equations to determine whether there are zero, one, or infinitely many solutions.
Answer
The only solution is $0$.
Answer for screen readers
The only solution is $0$.
Steps to Solve
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Distribute Terms We start by distributing the terms in the equation. Expand the left and right sides: $$ -10n + 3(8) + 3(8n) = -6(n) + 6(4) $$ which simplifies to: $$ -10n + 24 + 24n = -6n + 24 $$
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Combine Like Terms Now, combine like terms on both sides: Left side: $$ -10n + 24n + 24 = 14n + 24 $$ Right side: $$ -6n + 24 $$ So the equation now is: $$ 14n + 24 = -6n + 24 $$
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Isolate the Variable Next, we isolate the variable $n$. First, add $6n$ to both sides: $$ 14n + 6n + 24 = 24 $$ which simplifies to: $$ 20n + 24 = 24 $$ Now subtract 24 from both sides: $$ 20n = 0 $$
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Solve for n Now divide both sides by 20 to find $n$: $$ n = 0 $$
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Determine the Nature of Solutions Since we found a single solution for $n$, we conclude that the only solution is: $$ n = 0 $$
The only solution is $0$.
More Information
In this equation, we adjusted and simplified until we found a specific value for $n$. This process helps ensure that we understand how equations can provide either unique solutions or express certain conditions based on their structure.
Tips
- Sometimes, students forget to distribute correctly. Always double-check during this step.
- Not combining like terms properly can lead to incorrect results. Be mindful of similar terms!
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