Problem 12 - Algebra 2

Question

Solve each equation. Check your solution. $$ -\frac{3}{4} n+1=-11 $$

Step-by-Step Solution

Answer

The solution is \( n = 16 \).

Step 1: Isolate the variable term

Start by moving the constant from the left side to the right side of the equation. Subtract 1 from both sides to get the following equation:\[-\frac{3}{4} n + 1 - 1 = -11 - 1\]Simplifying both sides results in:\[-\frac{3}{4} n = -12\]

Step 2: Solve for the variable

Next, solve for \( n \) by eliminating the coefficient of \( n \). Multiply both sides of the equation by the reciprocal of \(-\frac{3}{4} \), which is \(-\frac{4}{3} \):\[-\frac{4}{3} \cdot \left(-\frac{3}{4} n\right) = -\frac{4}{3} \cdot (-12)\]Simplifying gives:\[n = 16\]

Step 3: Check the solution

Substitute \( n = 16 \) back into the original equation to verify:\[-\frac{3}{4} (16) + 1 = -11\]Simplifying the left side, calculate:\[-\frac{3}{4} \cdot 16 = -12\]Thus, the equation becomes:\[-12 + 1 = -11\]Since both sides equal \(-11\), the solution \( n = 16 \) is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that revolves around finding the value of a variable that makes the equation true. It involves manipulating the equation in a precise manner to isolate the variable on one side of the equal sign. Let’s break this down further.

• Identify the equation to solve: For example, consider \(-\frac{3}{4} n+1=-11\). The goal is to find the value of \(n\) that satisfies the equation.
• Isolate the variable: Start by moving constants to one side. Subtract \(1\) from both sides to focus on the term with \(n\).

Once the equation is simplified to \(-\frac{3}{4} n = -12\), the task is to eliminate the coefficient of the variable through further manipulation. Each step gets you closer to finding "n", ensuring clarity by maintaining balance on both sides of the equation.
Solving equations is a methodical process, much like solving a puzzle. Each move is carefully chosen to maintain the balance, eventually revealing the unknown.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or multiplication). They form the backbone of equations and can represent everything from simple arithmetic calculations to complex relationships. Let's break them down:

• Components: Each expression has numbers, commonly called coefficients, and variables. In \(-\frac{3}{4} n + 1\), \(-\frac{3}{4}\) is the coefficient of \(n\), and 1 is a constant.
• Simplification: The goal is often to combine like terms or simplify expressions to make equations easier to solve.

Understanding algebraic expressions is all about seeing these components and realizing how they fit together. They hold the key to forming and solving equations, turning a complex problem into manageable steps through simplification and organization.
Gaining fluency in recognizing and manipulating these expressions eases the means to solving equations accurately.

Inverse Operations

Inverse operations are operations that undo each other. They play a critical role in solving equations by allowing you to isolate variables.Understanding Inverse Operations:

• Addition and Subtraction: These are inverse operations. Subtracting \(1\) from \(\frac{3}{4} n + 1 = -11\) undoes the addition, helping isolate the term with the variable \(n\).
• Multiplication and Division: Also inverse operations. To solve \(-\frac{3}{4} n = -12\), you multiply both sides by \(-\frac{4}{3}\), effectively "undoing" the multiplication of \(-\frac{3}{4}\).

By employing inverse operations, what might seem like an analytical labyrinth becomes a straightforward path to the solution. They provide a reliable way to balance equations during the solving process, ensuring that each change keeps the equation valid.
Mastering inverse operations is not just about calculation but also understanding how to mediate between steps to maintain equation balance and lead to the correct result.