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Chapter 5: Problem 12

For Exercises 6 to \(32,\) solve and check. $$3 x+1=11-2 x$$

Short Answer

Expert verified
The value of \(x\) is \(2\).

Step by step solution

01

Rearrange the equation

Our first step is to bring all terms involving the variable on one side and all numerical terms on the opposite side. This means that we need to move \(2x\) from the right to the left side of the equation and -1 from the left to the right side of the equation. To do this, we add \(2x\) and subtract \(1\) from both sides of the equation: \(3x + 2x = 11 - 1\).
02

Simplify the equation

In the next step, we simplify both sides of the equation by calculating the expression values. The left side will become \(5x\) and the right side will become \(10\). Thus, our simplified equation is \(5x = 10\).
03

Solve for x

The final step is solving the equation for \(x\). This requires us to isolate \(x\) on one side of the equation. We do this by dividing both sides of the equation by \(5\): \(x = 10 / 5\).

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Key Concepts

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

Equation Rearrangement
Rearranging an equation is a fundamental step in solving linear equations. The goal is to move all terms that contain the variable of interest to one side of the equation and all constant numbers to the opposite side. This helps to create a clear path to finding the solution.
  • Identify which terms contain the variable you're solving for.
  • Move variable terms to one side of the equation. Add or subtract these terms on both sides to maintain balance.
  • Move constant terms to the opposite side of the equation by adding or subtracting them on both sides.
In our exercise, this involved moving the term \(2x\) from the right side to the left side and moving the numerical term \(-1\) to the right. This step gives us a more structured equation like \(3x + 2x = 11 - 1\). This rearrangement simplifies further steps.
Simplification
Simplification is the process of combining like terms and reducing expressions to their simplest form. In solving linear equations, simplification is vital to make the equation more manageable.
  • Combine like terms: Terms that have the same variable can be added or subtracted.
  • Simplify expressions: Calculate the numerical operations to reduce the equation.
For example, in our arranged equation \(3x + 2x = 11 - 1\), the left side combines to form \(5x\), and the right side simplifies to \(10\). This reduces the equation to \(5x = 10\), making it much clearer and easier to solve.
Solving for a Variable
The final step in solving a linear equation is isolating the variable of interest. Once the equation is rearranged and simplified, you solve for the variable by performing basic operations.
  • Isolate the variable: You need to have the variable on one side of the equation and the numerical terms on the other.
  • Use inverse operations: To isolate the variable, perform the opposite (inverse) operation to both sides.
In our example, the simplified equation is \(5x = 10\). To solve for \(x\), divide both sides by \(5\) to get \(x = 2\). At this point, you've solved the equation and found the value of the variable. This final step provides the solution we're searching for.

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Most popular questions from this chapter

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