Problem 49 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution is \(n = \frac{12}{7}\).

Step 1: Distribute and Simplify

Start by distributing the constants across the terms in parentheses.For \(-2(n-4)\): \[-2 \times n + (-2) \times (-4) = -2n + 8\]For \(-(3n-1)\): \[-3n + 1\]So the equation becomes:\[-2n + 8 - 3n + 1 = -2 + 2n - 1\]

Step 2: Combine Like Terms

Combine like terms on both sides of the equation.Left side:\[-2n - 3n + 8 + 1 = -5n + 9\]Right side:\[-2 - 1 + 2n = 2n - 3\]The equation is now:\[-5n + 9 = 2n - 3\]

Step 3: Move Variables to One Side

To get all the 'n' terms on one side, add \(5n\) to both sides:\[-5n + 9 + 5n = 2n - 3 + 5n\]This simplifies to:\[9 = 7n - 3\]

Step 4: Isolate the Variable

To isolate \(n\), add 3 to both sides:\[9 + 3 = 7n - 3 + 3\]This gives:\[12 = 7n\]

Step 5: Solve for n

Finally, divide both sides by 7 to solve for \(n\):\[n = \frac{12}{7}\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term outside a parenthesis by every term inside the parentheses. This is essential for simplifying equations, like the one given:

• A general form of the distributive property is: \( a(b + c) = ab + ac \)
• In our example, you'd apply it to: \(-2(n - 4)\) = \(-2n + 8\) and \(-(3n-1)\) = \(-3n + 1\).

This step breaks down the equation into individual components, making it easier to handle. Using the distributive property sets the stage for solving equations by removing parentheses and dealing directly with the terms.

Combining Like Terms

When dealing with an algebraic equation, combining like terms helps to simplify the equation further.Like terms are those that have the same variable raised to the same power. By combining them, you make the equation more manageable:

• For example, in our initial equation: \(-2n - 3n + 8 + 1\), the like terms \(-2n\) and \(-3n\) combine to form \(-5n\).
• On the other side: \(-2 + 2n - 1\) simplifies to \(2n - 3\).

By bringing these terms together, you tidy up the equation, leaving fewer terms to work with, and making it easier to solve.

Moving Terms Across the Equation

To solve an equation, you need to gather all variable terms on one side and constant terms on the other. This process is known as moving terms across the equation, and it's done by adding or subtracting terms from both sides.Here's how you apply this process:

• Original rearranged equation: \(-5n + 9 = 2n - 3\)
• Move all \(n\) terms to one side by adding \(5n\):\(9 = 7n - 3\)

This step organizes the equation into a form that is closer to the solution, setting up for isolating the variable.

Isolating the Variable

The final goal in solving linear equations is isolating the variable. This means you'll reorganize the equation to get the variable by itself on one side. Doing this allows you to identify the solution directly.Here are the steps we use in the example:

• Equation before isolating: \(9 = 7n - 3\)
• Add 3 to both sides to isolate terms with the variable: \(9 + 3 = 7n\) leading to \(12 = 7n\).
• Finally, divide both sides by 7 to solve for \(n\): \(n = \frac{12}{7}\).

These steps make it possible to solve for \(n\), giving a clear solution to the equation. This technique is crucial and widely used in algebraic computations.