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Chapter 2: Problem 99

Solve each equation, if possible. $$ 2 d+5=0 $$

Short Answer

Expert verified
The solution is \(d = -\frac{5}{2}\).

Step by step solution

01

Isolate the Variable Term

To solve the equation \(2d + 5 = 0\), our first step is to isolate the term containing \(d\). We do this by subtracting 5 from both sides of the equation:\[2d + 5 - 5 = 0 - 5\]This simplifies to:\[2d = -5\]
02

Solve for the Variable

Now that we have \(2d = -5\), we need to solve for \(d\) by getting \(d\) by itself. We divide both sides of the equation by 2:\[\frac{2d}{2} = \frac{-5}{2}\]This simplifies to:\[d = -\frac{5}{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving linear equations, our primary goal is to solve for the unknown variable. This often means isolating the variable on one side of the equation. In the exercise provided, the equation is \(2d + 5 = 0\). Our task is to isolate the term with the variable, which is the term involving \(d\). By performing the inverse operation, such as subtraction, we can effectively isolate the variable.
  • Identify the variable term, in this case, \(2d\).
  • Apply the inverse operation to eliminate additional terms; subtracting 5 from both sides helps us move towards isolating \(d\).
  • The goal is to have all terms that do not contain the variable on the opposite side of the equation.
This process simplifies the equation, leaving \(2d = -5\). Isolating the variable is critical because it sets the stage for further simplifications and ultimately finding the solution.
Solving Equations
Once variables are isolated, the next core step is solving the equation. Solving means finding the value of the unknown that makes the equation true. From the isolated form \(2d = -5\), the goal is to solve for \(d\). It's all about performing operations to simplify and balance the equation further.
  • Spot that the coefficient of \(d\) is 2.
  • Use division, the inverse operation of multiplication, to solve for \(d\).
  • Divide each side of the equation by 2 to solve \(d\).
This process yields \(d = -\frac{5}{2}\), which is the solution to the equation. Solving equations relies on understanding operations and their inverses to gradually unveil the value of the unknown.
Algebraic Manipulation
Algebraic manipulation involves the use of operations to rearrange and simplify equations. This concept is at the heart of what we do when solving equations. In our example, we used basic algebraic techniques to transform the equation, making it solvable.
  • Start by examining what operations will effectively simplify the equation.
  • Perform consistent operations on each side of the equation to maintain equality.
  • Understand how different algebraic rules apply, such as the distributive property, factoring, and combining like terms, even though they weren't needed here.
The manipulation allows us to express the variable clearly and concisely. In essence, algebraic manipulation gives us the tools and techniques needed to handle a variety of equations, making it possible to solve problems systematically and effectively.

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