Problem 18 - Prealgebra

Question

Solve each equation. Check your solution. $$2(3.5 n+6)=2.5 n-2$$

Step-by-Step Solution

Answer

The solution is \(n = -3.11\).

Step 1: Distribute on the Left Side

We start by distributing the 2 on the left side of the equation:\[ 2(3.5n + 6) = 2.5n - 2 \]This becomes:\[ 7n + 12 = 2.5n - 2 \]

Step 2: Move All Terms Involving n to One Side

Subtract \(2.5n\) from both sides to get all the \(n\)-terms together:\[ 7n - 2.5n + 12 = -2 \]This gives us:\[ 4.5n + 12 = -2 \]

Step 3: Move Constant Terms to the Other Side

Subtract 12 from both sides to isolate the \(n\)-term:\[ 4.5n = -2 - 12 \]Which results in:\[ 4.5n = -14 \]

Step 4: Solve for n

Divide both sides by 4.5 to solve for \(n\):\[ n = \frac{-14}{4.5} \]Simplifying the fraction yields:\[ n = -\frac{28}{9} \]So, \(n = -3.11\).

Step 5: Check the Solution

Substitute \(n = -3.11\) back into the original equation to verify:\[ 2(3.5(-3.11) + 6) = 2.5(-3.11) - 2 \]The left side calculates approximately to:\[ 2(-10.885 + 6) \approx 2 \times -4.885 = -9.77 \]And the right side calculates:\[ -7.775 - 2 = -9.775 \]Since both sides are approximately equal, the solution \(n = -3.11\) is verified.

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps simplify expressions by distributing a single term across terms inside parentheses. Let's apply it to the equation provided in the exercise. Start with the original expression \[ 2(3.5n + 6) \]The "2" outside the parentheses needs to be distributed to both terms inside—\(3.5n\) and \(6\). Follow these steps:

• Multiply the outer term by the first term inside the parentheses: \(2 \times 3.5n = 7n\).
• Then, multiply the outer term by the second term: \(2 \times 6 = 12\).

The resulting expression is \[ 7n + 12 \]This transformation of the left side of the equation will make it easier for us to solve for \(n\), which we'll do in the next steps.

Isolating Variables

To isolate variables means to get the variable of interest by itself on one side of the equation. This is an essential step in solving any algebraic equation. For the given equation \[7n + 12 = 2.5n - 2\]the goal is to have all terms involving "n" on one side. Start by subtracting \(2.5n\) from both sides:

• This results in \(7n - 2.5n = 4.5n\).
• Thus, you have \(4.5n + 12 = -2\).

Once you have combined like terms (the terms with \(n\)), the variable "n" is now isolated on the left side. Next, remove the constant term \(12\) by subtracting it from both sides:\[ 4.5n + 12 - 12 = -2 - 12 \]Simplifying this gives:\[ 4.5n = -14 \]This prepares you for the final step of solving for \(n\).

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In the equation we've derived:\[ 4.5n = -14 \]we need to solve for "n." This process involves dividing both sides by the coefficient of the variable, which in this case is "4.5."

• Divide each side by 4.5: \(n = \frac{-14}{4.5}\).
• Simplify the fraction: \(n = -\frac{28}{9}\).
• Finally, convert the fraction to a decimal (if necessary): \(n \approx -3.11\).

Now, you've found that \(n = -3.11\), which should be verified in the next step to ensure it's the correct solution.

Checking Solutions

After solving an equation, it's always a good idea to check the solution by substituting it back into the original equation. Let's verify the solution \(n = -3.11\). Substitute this back into the original left-side expression:\[2(3.5(-3.11) + 6)\]Perform the calculations:

• First, compute inside the parentheses: \(3.5 \times -3.11 + 6 = -10.885 + 6 = -4.885\).
• Then, multiply by 2: \(2 \times -4.885 \approx -9.77\).

Now, compute the right side of the original equation:\[2.5(-3.11) - 2\]

• Multiply: \(2.5 \times -3.11 = -7.775\).
• Subtract 2: \(-7.775 - 2 = -9.775\).

Both sides are approximately equal, validating that the solution \(n = -3.11\) satisfies the original equation.