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Chapter 2: Problem 70

\(4 x-6+3 x-6 x=3+10\)

Short Answer

Expert verified
x = 19

Step by step solution

01

Combine Like Terms

First, combine the like terms on the left side of the equation. The terms are: \(4x\), \(3x\), and \(-6x\).\(4x + 3x - 6x - 6 = 3 + 10\)This simplifies to:\(x - 6 = 13\)
02

Isolate the Variable

Next, isolate the variable \(x\) by adding 6 to both sides of the equation:\(x - 6 + 6 = 13 + 6\)This simplifies to:\(x = 19\)

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Key Concepts

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

Combining Like Terms
In equations like the one given, the first step is to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, in our exercise, we have the terms: \(4x, 3x\), and \(-6x\).

To combine these:
  • Add or subtract the coefficients (the numbers in front of the variables).

So, \(4x + 3x = 7x\) and then \(7x - 6x = x\). This helps to simplify the expression on one side of the equation.

After combining the like terms in the given problem, we get \(x - 6 = 13\). This dramatically cuts down on the complexity of the equation and brings us closer to finding the variable.
Isolating the Variable
Once like terms have been combined, the goal is to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. This allows us to find its value.

  • To isolate the variable, we perform inverse operations. Inverse operations are operations that 'undo' each other, like addition and subtraction.

Starting with our simplified equation \(x - 6 = 13\), we add 6 to both sides. This operation cancels out the -6 on the left side and shifts it to the right side.

\(x - 6 + 6 = 13 + 6\) simplifying to \(x = 19\). Now the variable \(x\) is isolated, giving us the solution.
Simplification
Simplification is crucial in solving linear equations. It involves breaking down complicated expressions into simpler forms.

  • This process includes combining like terms and isolating the variable but also involves making sure every step follows logical rules.

Initially, the equation \(4x - 6 + 3x - 6x = 3 + 10\) can seem complex. Through simplification, we first combine like terms to get \(x - 6 = 13\).

Then, isolate the variable to find \(x = 19\). These steps ensure the equation is as straightforward as possible, making it easier to solve. Remember, each small simplification brings you closer to the solution, making the overall problem-solving process efficient.

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