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Chapter 2: Problem 78

Solve each equation, if possible. $$ -2 r+8=-1 $$

Short Answer

Expert verified
The solution is \( r = \frac{9}{2} \) or \( r = 4.5 \).

Step by step solution

01

Isolate the Variable Term

Start by getting the term with the unknown, \( r \), on one side by subtracting 8 from both sides of the equation: \(-2r+8-8=-1-8\). This simplifies to \-2r = -9\.
02

Solve for the Variable

Now, solve for \( r \) by dividing both sides of the equation by -2: \( \frac{-2r}{-2} = \frac{-9}{-2} \). Simplifying, this becomes \( r = \frac{9}{2} \) or \( r = 4.5 \).

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Key Concepts

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

Isolating Variables
Solving linear equations often begins with isolating the variable. This means getting the variable you want to solve for, like \( r \) in our example, all by itself on one side of the equation.
To start, identify the terms that are keeping the variable from being isolated. In our case, the variable term is \(-2r\), and it's accompanied by a constant term, \(+8\).
The goal is to remove these constant terms, or any other variables, that are added or subtracted from your target variable.
  • In the given equation \(-2r + 8 = -1\), the constant \(+8\) is subtracted from both sides to move it away from the \( r \) term.
  • After subtracting, the equation then reads \(-2r = -9\).
This key step allows us to tackle the variable by itself, ready to be solved.
Elementary Algebra
Elementary Algebra is like a toolbox for solving basic equations. It's filled with techniques that allow you to manipulate equations to find unknown values. Two of the most fundamental tools are addition/subtraction and multiplication/division.
They help change an equation's form without altering its value.
  • Addition and subtraction are inverse operations that help balance equations. In our example, subtracting 8 from both sides keeps the equation balanced while shifting the constant.
  • Multiplication and division are used to "undo" multiplication or division associated with variables. For \(-2r = -9\), dividing both sides by \(-2\) cancels out the coefficient of the variable.
Keeping these operations in your toolbox will assist with various algebraic problems. Understanding when and how to apply them is essential.
Equation Simplification
Equation simplification is about making an equation easier to work with, by combining like terms and reducing the number of elements involved. For a simpler path to the solution:
First, look to remove any extra numbers or terms around your variable.
After isolating the variable term such as \(-2r = -9\), the next step is to simplify further.
  • Divide each side by the coefficient of the variable, in this case, \(-2\).
  • This action gives us \( r = \frac{9}{2} \) or \( r = 4.5 \), turning a complex equation into a straightforward solution.
Simplifying equations is about recognizing opportunities to condense or rephrase until the solution reveals itself. Practicing these steps makes complex equations much more approachable.

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