Problem 45 - Algebra for College Students

Question

Solve each equation. \(-2(3 n-1)+3(n+5)=-4(n-4)\)

Step-by-Step Solution

Answer

The solution is \(n = -1\).

Step 1: Distribute

First, distribute the constants into the parentheses on both sides of the equation:\[-2(3n-1) = -2 \cdot 3n + (-2) \cdot (-1) = -6n + 2\]\[3(n+5) = 3n + 15\]Thus, the equation becomes:\[-6n + 2 + 3n + 15 = -4(n - 4)\]

Step 2: Simplify

Combine like terms on the left side of the equation:\[-6n + 3n + 2 + 15 = -3n + 17\]Thus, the equation now is:\[-3n + 17 = -4(n - 4)\]

Step 3: Distribute on the Right Side

Distribute the \(-4\) on the right side of the equation:\[-4(n-4) = -4 \cdot n + (-4) \cdot (-4) = -4n + 16\]So, the equation now is:\[-3n + 17 = -4n + 16\]

Step 4: Solve for n

Isolate \(n\) by adding \(4n\) to both sides of the equation:\[-3n + 4n + 17 = 4n - 4n + 16\]This simplifies to:\[n + 17 = 16\]Then, subtract 17 from both sides:\[n = 16 - 17\]\[n = -1\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that involves multiplying a single term by each term inside parentheses. It helps simplify expressions so you can solve them more easily. In this particular exercise, the distributive property is used to "distribute" numbers across terms inside the parentheses, transforming it into a simpler form that can be solved step by step. For example:

• When distributing \(-2(3n - 1)\): Multiply \(-2\) by \(3n\) and \(-1\), resulting in \(-6n + 2\).
• For \(3(n+5)\): Distribute \(3\) to both \(n\) and \(5\), which gives \(3n + 15\).
• On the right side, \(-4(n-4)\) distributes to give \(-4n + 16\).

Using the distributive property is crucial for breaking down equations, making each term clearer and allowing for further simplification in the next steps.

Combining Like Terms

Once you have distributed, the next key step is combining like terms. Like terms are terms that contain the same variable raised to the same power. This step simplifies the expression by consolidating similar terms, which reduces the equation's complexity. Here's how we did it with our given equation:

• The initial expression \(-6n + 2 + 3n + 15\) contains the terms \(-6n\) and \(3n\) as like terms. Combining them results in \(-3n\).
• Similarly, the constants \(2\) and \(15\) can be combined to make \(17\).

Thus, the expression becomes \(-3n + 17\). The right side of the equation also relies on combining distributed terms to match patterns on both sides, as seen with \(-4(n-4)\) which also simplifies to \(-4n + 16\). Recognizing and merging like terms efficiently reduces confusion and sets the stage to isolate the desired variable.

Solving for a Variable

The final goal of solving algebraic equations is to find the value of the variable, which often requires isolating that variable on one side of the equation. In this case, the variable is \(n\). After distributing and combining like terms:

• We have \(-3n + 17\) on the left and \(-4n + 16\) on the right.
• To isolate \(n\), move terms involving \(n\) to one side. Adding \(4n\) to both sides gives us \(n + 17 = 16\).
• Finally, solve for \(n\) by subtracting \(17\) from both sides, resulting in \(n = -1\).

The steps illustrate how systematic movements of terms enable us to find the value of the variable \(n\). Isolation allows for clear interpretation of the solution, ensuring each side of the equation stays balanced until the final result is achieved.