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Chapter 2: Problem 56

\(13 t+9=12 t+9\)

Short Answer

Expert verified
t = 0

Step by step solution

01

Set up the Equation

Look at the given equation. We have: \[ 13t + 9 = 12t + 9 \]
02

Subtract 12t from Both Sides

To isolate the variable t, we first need to eliminate one of the t terms. Subtract 12t from both sides of the equation: \[ 13t - 12t + 9 = 12t - 12t + 9 \] Simplifying this, we get: \[ t + 9 = 9 \]
03

Subtract 9 from Both Sides

Next, to completely isolate t, subtract 9 from both sides of the equation: \[ t + 9 - 9 = 9 - 9 \] Simplifying this, we get: \[ t = 0 \]

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Key Concepts

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

Isolating Variable
In solving linear equations, a key step is isolating the variable. This means rearranging the equation so that the variable appears alone on one side of the equation. Understanding this concept is crucial because it transforms a complex equation into one that is simple to solve.
To isolate a variable, you can:
  • Add or subtract terms on both sides of the equation
  • Multiply or divide both sides by a constant
For instance, consider the equation \[ 13t + 9 = 12t + 9 \] Our goal is to isolate the variable \(t\). By following a systematic approach and rearranging the terms, we simplify the process of finding the solution.
Subtracting Terms
Subtracting terms is a fundamental process in solving equations. It's often used to move terms from one side of the equation to the other.
For example, to isolate \(t\) in \[ 13t + 9 = 12t + 9\],
we first subtract \(12t\) from both sides: \[13t - 12t + 9 = 12t - 12t + 9\] which simplifies to: \[t + 9 = 9\].
Here, subtracting 12t from both sides helps in making the equation simpler while trying not to change its balance. It's important to perform the same operation on both sides to maintain equality. This step is crucial because it leads us to a simpler equation where isolating the variable becomes straightforward.
Simplifying Equations
Simplifying equations involves combining like terms and performing basic arithmetic operations to make the equation easier to work with.
Consider our simplified equation: \[ t + 9 = 9 \].
To further isolate \(t\), subtract 9 from both sides: \[ t + 9 - 9 = 9 - 9 \]. Simplifying this, we get: \[ t = 0 \].
Simplifying makes the equation easier to handle and solve. By performing proper operations like addition, subtraction, multiplication, or division, you reduce the equation to its simplest form. This step ensures clarity and ease when finding the solution.

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