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Chapter 2: Problem 67

\(3 x+6-10=2 x-2\)

Short Answer

Expert verified
x = 2

Step by step solution

01

Isolate variable terms on one side

Subtract 2x from both sides of the equation to start isolating the variable terms: 3x + 6 - 10 - 2x = 2x - 2 - 2x
02

Simplify the equation

Combine like terms to simplify both sides: (3x - 2x) + (6 - 10) = -2 x - 4 = -2
03

Solve for x

Add 4 to both sides to isolate x: x - 4 + 4 = -2 + 4 x = 2

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Key Concepts

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

isolating variables
Isolating variables is one of the fundamental steps in solving linear equations. It means getting the variable (like x) by itself on one side of the equation.
In our given problem, the first step involves isolating the variable terms. Let's break it down:
Start with the equation: \[3x + 6 - 10 = 2x - 2\]
To isolate the variable terms, subtract 2x from both sides:
\[3x + 6 - 10 - 2x = 2x - 2 - 2x\]
This step ensures we get rid of the x from the right side, making it easier to focus on solving for x on one side only. Remember, whatever you do on one side of the equation, you must do to the other side too!
simplification
Simplification is the process of making an equation easier to solve. Simplifying the expression involves combining like terms and performing arithmetic operations.
In our equation, after isolating the variable, we have: \[(3x - 2x) + (6 - 10) = -2\]
Breaking it down further: \[1x - 4 = -2\]
Combining like terms such as 3x and -2x simplifies the variables on the left side. Similarly, combining 6 and -10 simplifies the constants. As a result, we have a much simpler equation that we can work with more easily.
combining like terms
Combining like terms refers to merging terms that have the same variable raised to the same power. This step is crucial to simplify the equation before further solving.
Looking at our simplified equation: \[(3x - 2x) + (6 - 10) = -2\]
We can see that: \[3x - 2x = x\] and \[6 - 10 = -4\]
Thus, the equation becomes: \[ x - 4 = -2\]
By combining like terms, we make the equation easier to solve. It鈥檚 essential to ensure the variable terms and constant terms are correctly merged.
algebraic manipulation
Algebraic manipulation involves using algebra rules and operations to modify the equation to find the solution.
In the final step of our problem, we need to isolate x completely: \[x - 4 = -2\]
To do this, we add 4 to both sides of the equation:
\[x - 4 + 4 = -2 + 4\]
This results in: \[x = 2\]
Through algebraic manipulation, we have isolated x and found that its value is 2. This demonstrates how using basic operations like addition helps in isolating the variable to solve the equation.

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