Problem 33 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 3(n-5)-(6-2 n)=4 n $$

Step-by-Step Solution

Answer

The solution is \(n = 21\).

Step 1: Distribute

First, we need to distribute the numbers outside the parentheses to simplify both sides of the equation. For the left side, distribute 3: \[ 3(n-5) = 3n - 15 \] For the expression \(-(6-2n)\):\[ -(6-2n) = -6 + 2n \] So, the equation becomes: \[ 3n - 15 - 6 + 2n = 4n \]

Step 2: Simplify Both Sides

Combine like terms on the left side of the equation:\[ (3n + 2n) - 21 = 4n \]which simplifies to:\[ 5n - 21 = 4n \]

Step 3: Isolate the Variable n

Subtract \(4n\) from both sides to move all terms with \(n\) to one side:\[ 5n - 4n - 21 = 0 \]Simplify:\[ n - 21 = 0 \]

Step 4: Solve for n

Add 21 to both sides to solve for \(n\):\[ n = 21 \]

Step 5: Check the Solution

Substitute \(n = 21\) back into the original equation to verify:Original equation:\[ 3(n-5)-(6-2n)=4n \]Substitute \(n = 21\):Left side:\[ 3(21-5) - (6-2(21)) = 3(16) - (6-42) = 48 - (-36) = 48 + 36 = 84 \]Right side:\[ 4(21) = 84 \]Both sides equal 84, so \(n = 21\) is correct.

Key Concepts

Simplifying Expressions

Simplifying expressions is one of the foundational steps in solving algebraic equations. It involves breaking down complex expressions into simpler, more manageable parts. This process often includes distributing and applying basic arithmetic operations.

Distributing refers to multiplying a single term with terms inside parentheses. For example, in the expression \(3(n-5)\), you multiply 3 by both \(n\) and -5, resulting in \(3n - 15\). Another example is \(-(6-2n)\), where you distribute -1 across the terms in parentheses to get \(-6 + 2n\).

The main goal of simplifying is to create expressions that are easier to work with when solving equations. Once an expression is simplified, it sets the stage for the next steps in the problem-solving process.

Combining Like Terms

After simplifying expressions through methods such as distributing, the next step is to combine like terms. This involves adding or subtracting terms with the same variables and exponents.

In our example, the terms \(3n\) and \(2n\) on the left side of the equation are like terms. You can combine these to get \(5n\). Additionally, you can combine constant terms such as \(-15\) and \(-6\) to get \(-21\).

Combining like terms helps to further condense the equation. The equation \(3n + 2n - 21 = 4n\) becomes \(5n - 21 = 4n\) once like terms are combined. This simplification makes the equation easier to solve by creating fewer terms to manipulate.

Solving for Variables

Once the equation is simplified and like terms are combined, the next step is solving for the variable. The goal is to isolate the variable on one side of the equation to determine its value.

In the provided equation, we move all terms containing \(n\) to one side by subtracting \(4n\) from both sides: \(5n - 4n - 21 = 0\), which simplifies to \(n - 21 = 0\). After isolating the \(n\) term, you solve for \(n\) by performing basic arithmetic, such as adding the constant on both sides: \(n = 21\).

This isolation and solving process helps transition from a complex equation to a straightforward solution. The clearer the path to the variable, the more intuitive solving becomes.

Checking Solutions in Equations

To ensure correctness, it's crucial to check the solution in the original equation. This step confirms that the value found for the variable makes both sides of the equation equal.

By substituting \(n = 21\) back into the original problem, we verify each part matches: - The left side, \(3(21-5) - (6-2(21))\), calculates to \(16\cdot3\) plus \(-36\), equaling 84.- The right side computes to \(21\cdot4\), which is also 84.

Since both sides equate to the same value, \(n = 21\) is confirmed as correct. This checking step is vital, as it acts as a safeguard against errors during solving and provides confidence in the solution.