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Chapter 2: Problem 21

\(t+12.3=-4.6\)

Short Answer

Expert verified
The solution is t = -16.9.

Step by step solution

01

Understand the Equation

The equation provided is a simple linear equation: \[ t + 12.3 = -4.6 \]The goal is to isolate the variable \( t \).
02

Subtract 12.3 from Both Sides

To isolate \( t \), subtract 12.3 from both sides of the equation:\[ t + 12.3 - 12.3 = -4.6 - 12.3 \]This simplifies to:\[ t = -16.9 \]
03

Verify the Solution

To verify, substitute \( -16.9 \) back into the original equation:\[ -16.9 + 12.3 = -4.6 \]Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Isolating Variables
In solving linear equations, isolating the variable is a crucial step. A linear equation like \( t + 12.3 = -4.6 \) contains a variable (\( t \)) that we need to solve for. To isolate the variable means to get the variable (in this case, \( t \)) alone on one side of the equation.

In this exercise, we did this by simplifying the equation through subtraction. Here's how it works:
  • Take \( t + 12.3 = -4.6 \)
  • Subtract 12.3 from both sides \[ t + 12.3 - 12.3 = -4.6 - 12.3 \]
  • This simplifies to \[ t = -16.9 \]


Now we have isolated \( t \) on one side, and the equation tells us that \( t = -16.9 \). Isolating the variable like this makes it easy to find its value.
Verification of Solutions
Verification of solutions is the process of checking that the solution obtained is correct. After isolating the variable and solving for it, we need to substitute the solution back into the original equation to ensure it holds true.

Let's verify our solution for \( t = -16.9 \). We start with the original equation: \[ t + 12.3 = -4.6 \].
  • Substitute \( -16.9 \) for \( t \): \[ -16.9 + 12.3 = -4.6 \]
  • Simplify the left-hand side: \[ -4.6 = -4.6 \]


Both sides of the equation are indeed equal. This confirms that \( t = -16.9 \) is a valid solution, as it satisfies the original equation.
Linear Equations
Linear equations are the foundation of algebra and an essential concept in mathematics. A linear equation is an equation in which the highest power of the variable is one.

The general form of a linear equation is:
\[ ax + b = c \]

where \( a \), \( b \), and \( c \) are constants. In simpler terms, a linear equation graphs as a straight line, hence the name 'linear'.
  • The key to solving linear equations involves isolating the variable term and performing basic arithmetic operations like addition, subtraction, multiplication, or division.
  • In our exercise, \[ t + 12.3 = -4.6 \] is a linear equation. The solution involves straightforward steps utilizing the principles of isolating variables and verification of solutions.


Understanding linear equations not only helps solve individual problems but also builds a strong foundation for more advanced mathematical concepts.

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