Problem 22 - Elementary Algebra

Question

Solve. $$ 2 n+13=-35 $$

Step-by-Step Solution

Answer

n = -24

Step 1: Isolate the Term with the Variable

To solve the equation \(2n + 13 = -35\), we first need to isolate the term with the variable \(n\). Subtract 13 from both sides of the equation:\[2n + 13 - 13 = -35 - 13\]This simplifies to:\[2n = -48\]

Step 2: Solve for the Variable

Now that we have \(2n = -48\), divide both sides of the equation by 2 to solve for \(n\):\[\frac{2n}{2} = \frac{-48}{2}\]This gives us:\[n = -24\]

Key Concepts

Understanding Algebra in Equations

Algebra is a powerful branch of mathematics that deals with symbols and rules for manipulating those symbols. In most cases, these symbols represent quantities without necessarily specifying what the quantities are. This versatility makes algebra an essential tool for solving various types of equations.
In an equation like \(2n + 13 = -35\), algebra helps to describe a relationship between numbers, using the variable \(n\) to represent an unknown number. The goal when solving such problems is to find the value of this unknown variable.

• The equation \(2n + 13 = -35\) consists of an expression, \(2n + 13\), set equal to a number, \(-35\).
• Understanding algebra is key to manipulating the expressions and isolating the variable to solve for it.

With algebra, we can systematically approach and solve equations using clear, logical steps.

Steps in Solving Equations

When solving equations, it's important to follow a series of steps that lead you to the solution. The process involves simplifying equations and performing operations to both sides to maintain equality.
For the equation \(2n + 13 = -35\), the solving process consists of key steps:

• Identify and Simplify: Simplify complex expressions on both sides of the equation if needed. In this example, the expression \(2n + 13\) is already simplified.
• Remove Additive Constants: To isolate the variable terms, subtract any constants added to the variable. Subtracting 13 from both sides changes the equation to \(2n = -48\).
• Isolate the Variable: Finally, divide both sides of the equation by the coefficient of the variable to make the variable stand alone. Dividing by 2 results in \(n = -24\).

These steps ensure that you systematically break down equations into simpler forms while keeping the equation balanced until the solution is found.

Variable Isolation Techniques

Variable isolation is the process of rearranging an equation so that the variable of interest is on one side by itself. Achieving this involves a range of algebraic techniques, mainly aimed at undoing operations applied to the variable.

• Undo Addition or Subtraction: As shown in the exercise, if a constant is added to the term with the variable, subtract it from both sides. So, \(2n + 13 = -35\) becomes \(2n = -48\).
• Undo Multiplication or Division: If the variable term is multiplied by a number (in our case, \(2n\)), divide both sides by that number to isolate \(n\). Hence, \(2n = -48\) becomes \(n = -24\).

This method of variable isolation helps ensure each step remains balanced, meaning what is done to one side of the equation must also be done to the other. This balance is crucial to maintaining the integrity of the equation throughout the solving process.