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Chapter 2: Problem 25

Solve each equation, and check your solution. $$ x-8.4=-2.1 $$

Short Answer

Expert verified
x = 6.3

Step by step solution

01

Understand the equation

The given equation is written in the form of a linear equation. The equation is: x - 8.4 = -2.1
02

Isolate the variable

To isolate x, add 8.4 to both sides of the equation. This cancels out -8.4 on the left side. x - 8.4 + 8.4 = -2.1 + 8.4 x = 6.3
03

Check the solution

To verify the solution, substitute x with 6.3 in the original equation. 6.3 - 8.4 = -2.1 Since -2.1 = -2.1 is a true statement, the solution is correct.

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra. These equations represent a straight line when plotted on a graph. The standard form for a linear equation in one variable is \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. The goal in linear equations is often to find the value of the variable that makes the equation true. Understanding and solving linear equations is crucial for more advanced topics in math, like calculus and linear algebra.
solving equations
Solving equations involves finding the values of the variables that make the equation true. To solve the equation \( x - 8.4 = -2.1 \), you need to get \( x \) by itself on one side of the equation. This process involves performing operations that keep the equation balanced. Always do the same thing to both sides. In our example, adding 8.4 to both sides helps achieve this balance, leading to the equation \( x = 6.3 \).
isolation of variables
Isolation of variables is a key step in solving equations. The objective is to get the variable by itself on one side of the equation. For the equation \( x - 8.4 = -2.1 \), you isolate \( x \) by adding 8.4 to both sides:

\[ x - 8.4 + 8.4 = -2.1 + 8.4 \]
This simplifies to:
\[ x = 6.3 \]
By performing the same operation on both sides, the equation remains balanced, and you get the value of \( x \).
checking solutions
Checking your solution is an important step to ensure accuracy. Once you find that \( x = 6.3 \), substitute it back into the original equation to verify.
Substitute \( x \):
\[ 6.3 - 8.4 = -2.1 \]
Since this simplifies to \( -2.1 = -2.1 \), the solution is correct. Always check your work to avoid errors, especially in more complex equations.
basic algebra
Basic algebra embodies skills like simplifying expressions, solving equations, and understanding variables and constants. With linear equations like \( x - 8.4 = -2.1 \), you apply these foundational skills.
Key steps include:
  • Identifying the parts of the equation
  • Performing operations to isolate the variable
  • Checking your solutions to ensure accuracy
Mastering these fundamental techniques ensures a strong foundation for advanced mathematical topics.

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