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Magazine Chapter 8: Problem 13
Solve each equation. Check your solution. $$6 n-18=4(n+2.1)$$
Short Answer
Expert verified
The solution is \(n = 13.2\).
Step by step solution
01
Distribute the 4 on the Right
To solve the equation \(6n - 18 = 4(n + 2.1)\), first distribute the 4 to both terms inside the parentheses on the right side. This gives you: \(6n - 18 = 4n + 8.4\).
02
Move Variables to One Side
Subtract \(4n\) from both sides of the equation to get the terms with \(n\) on one side: \(6n - 4n - 18 = 8.4\), simplifying to \(2n - 18 = 8.4\).
03
Isolate the Variable Term
Add 18 to both sides to isolate the \(n\) term: \(2n - 18 + 18 = 8.4 + 18\). This simplifies to \(2n = 26.4\).
04
Solve for n
Divide both sides by 2 to solve for \(n\): \(n = \frac{26.4}{2}\), resulting in \(n = 13.2\).
05
Check the Solution
Substitute \(n = 13.2\) back into the original equation to check if it holds true: \(6(13.2) - 18 = 4(13.2 + 2.1)\). Calculate each side separately: \(6(13.2) - 18 = 79.2 - 18 = 61.2\) and \(4(13.2 + 2.1) = 4(15.3) = 61.2\). Both sides equal, confirming the solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equation Solving
Equation solving is a process of finding the value of a variable that makes an equation true. At its core, it involves ensuring that both sides of the equation balance or hold the same value.
For example, in the equation \(6n - 18 = 4(n + 2.1)\), our goal is to determine the value of \(n\) that makes both sides equal.
To solve such equations, one performs a series of algebraic manipulations: simplifying expressions, rearranging terms, and undoing operations through inverse operations, like adding to undo subtraction or dividing to undo multiplication.
For example, in the equation \(6n - 18 = 4(n + 2.1)\), our goal is to determine the value of \(n\) that makes both sides equal.
To solve such equations, one performs a series of algebraic manipulations: simplifying expressions, rearranging terms, and undoing operations through inverse operations, like adding to undo subtraction or dividing to undo multiplication.
Distributive Property
The distributive property is a fundamental algebraic property used often while solving equations. It states that multiplication distributed over addition acts as: \(a(b + c) = ab + ac\).
This property is crucial when dealing with terms inside parentheses. Especially when you have an equation like \(6n - 18 = 4(n + 2.1)\).
This property is crucial when dealing with terms inside parentheses. Especially when you have an equation like \(6n - 18 = 4(n + 2.1)\).
- Here, you distribute the 4 across each term inside the parentheses: distribute it to \(n\) and then to 2.1.
- This simplifies the equation to \(4n + 8.4\).
Isolate the Variable
Isolating the variable means manipulating the equation in such a way that one side contains only the variable you are solving for.
In our example, once we applied the distributive property resulting in \(6n - 18 = 4n + 8.4\), the next steps are crucial:
In our example, once we applied the distributive property resulting in \(6n - 18 = 4n + 8.4\), the next steps are crucial:
- First, get all \(n\) terms on one side by subtracting \(4n\) from both sides, simplifying to \(2n - 18 = 8.4\).
- Then, use addition to move the constant term (-18) to the other side by adding 18, resulting in \(2n = 26.4\).
- Lastly, divide by the coefficient of \(n\), which is 2, to finally solve for \(n\), getting \(n = 13.2\).
Check the Solution
Checking the solution is an essential step in solving equations. It confirms whether the value obtained for the variable satisfies the original equation.
After finding \(n = 13.2\), substitute it back into the original equation: \(6(13.2) - 18 = 4(13.2 + 2.1)\).
After finding \(n = 13.2\), substitute it back into the original equation: \(6(13.2) - 18 = 4(13.2 + 2.1)\).
- Calculate each side separately to ensure they are equal: the left side calculates to \(61.2\) and so does the right side.
- The fact that both sides equal confirms our solution \(n = 13.2\) is indeed correct.
