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Chapter 2: Problem 82

Solve equation. \(2 x+5 x-3 x+4=3 x+2\)

Short Answer

Expert verified
x = -2

Step by step solution

01

- Combine like terms on the left side

Combine the terms with the variable x: \(2x + 5x - 3x + 4 = 3x + 2\) This simplifies to:\(4x + 4 = 3x + 2\)
02

- Get all x terms on one side

Subtract \(3x\) from both sides to get all the x terms on the left side:\(4x + 4 - 3x = 3x + 2 - 3x\)This simplifies to:\(x + 4 = 2\)
03

- Isolate the variable x

Subtract 4 from both sides to isolate the variable x:\(x + 4 - 4 = 2 - 4\)This simplifies to:\(x = -2\)

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Key Concepts

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

Combining Like Terms
When solving linear equations, combining like terms is crucial. Like terms are terms that have the same variable raised to the same power. For example, in the equation provided, we have these terms: \(2x\), \(5x\), and \(-3x\).
To simplify, sum the coefficients (numbers in front of the variables). Here, add \(2x + 5x\), which gives \(7x\). Then, subtract \(3x\) from \(7x\), leading to \(4x\). Our equation now looks simpler: \(4x + 4 = 3x + 2\). By combining like terms, you reduce complexity and make the equation easier to solve.
Remember, this technique helps in reassembling the equation into a neater and more understandable form, setting the stage for isolating the variable.
Isolating Variables
After combining like terms, the next step is isolating the variable. This means getting the variable on one side of the equation and the numbers on the other. To do this efficiently:
  • Subtract or add terms to both sides
  • Use inverse operations
From our simplified equation, \(4x + 4 = 3x + 2\), subtract \(3x\) from both sides. This leads us to \(x + 4 = 2\), further simplifying our equation.
Next, we need \(x\) alone. Subtract \(4\) from both sides, isolating \(x\): \(x + 4 - 4 = 2 - 4\) giving us \(x = -2\). Successfully isolating the variable \(x\) gives us the solution for the initial equation.
Basic Algebra
Having strong foundational skills in basic algebra is essential for solving linear equations. Basic algebra involves understanding and applying rules and operations on algebraic expressions.
  • Addition and subtraction of terms
  • Multiplication and division of terms
In our example, we employed basic algebra techniques like combining like terms and isolating variables. Familiarity with these operations allows you to approach and solve equations systematically. Solving linear equations is a fundamental algebraic skill that is built upon these simple, yet powerful techniques.
Keep practicing these basic skills to enhance your confidence and ability to solve more complex equations.

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