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Magazine Chapter 2: Problem 49
Solve. See Examples 1 through 7 $$ 4(2 n+1)=3(6 n+3)+1 $$
Short Answer
Expert verified
The solution is \( n = -\frac{3}{5} \).
Step by step solution
01
Distribute the Constants
First, distribute the constants across each bracket on both sides of the equation.For the left side: \[ 4(2n + 1) = 8n + 4 \]For the right side:\[ 3(6n + 3) = 18n + 9 \]The equation becomes: \[ 8n + 4 = 18n + 9 + 1 \] which simplifies to \[ 8n + 4 = 18n + 10 \].
02
Move Variables to One Side
To isolate the variable terms to one side, subtract \(8n\) from both sides:\[ 8n + 4 - 8n = 18n + 10 - 8n \]This gives us:\[ 4 = 10n + 10 \].
03
Move Constants to the Other Side
Subtract 10 from both sides to isolate the term with the variable on the right side:\[ 4 - 10 = 10n + 10 - 10 \]This simplifies to:\[ -6 = 10n \].
04
Solve for the Variable
Divide both sides of the equation by 10 to solve for \(n\):\[ \frac{-6}{10} = n \]Simplifying the fraction gives us:\[ n = -\frac{3}{5} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributing Constants
Distributing constants is a key step in solving algebraic equations that involve parentheses. When you see an equation like the original exercise, where a constant is multiplied by a polynomial inside parentheses, you need to apply the distributive property.
- The distributive property states that the multiplication of a constant and the sum within parentheses can be rewritten as the sum of each term in the parentheses multiplied by the constant.
- In our exercise, we see this in action as: \(4(2n + 1)\) becomes \(8n + 4\).
- Similarly, for the right side, \(3(6n + 3)\) is simplified to \(18n + 9\).
Isolating Variables
The process of isolating variables involves rearranging the equation to get the variable on one side. This is a crucial step in solving linear equations as it allows us to focus on the variable we're trying to solve for.
- First, ensure all terms containing the variable are on one side of the equation. In our example, subtracting \(8n\) from both sides does this, giving us \(4 = 10n + 10\).
- Next, you need to move constants (numbers without variables) to the opposite side. Here, we subtract 10 from both sides, simplifying to \(-6 = 10n\).
Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form. Once you solve the variable equation and are left with a fraction, taking this step is essential to convey a cleaner, more precise answer.
- Look at the fraction's numerator and denominator to find any common factors. In the solution example, \(-\frac{6}{10}\) can be simplified.
- Both 6 and 10 are divisible by 2, allowing us to simplify the fraction: \(-\frac{6}{10} = -\frac{3}{5}\).
Solving Linear Equations
Solving linear equations involves a few systematic steps, much like the ones we've walked through. This process helps balance equations to eventually find the value of the unknown variable.
- Start by distributing constants, as we've explored. This simplifies the expression and makes it more manageable.
- Next, move on to isolate the variable by ensuring all terms with the variable are on one side and constants on the other.
- Once isolated, if the variable is multiplied by a coefficient, divide both sides by this coefficient to solve for the variable.
- Finally, ensure any fraction resulting from the division is simplified.
