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Chapter 4: Problem 128

Solve \(2 x-5=7 x\)

Short Answer

Expert verified
x = -1

Step by step solution

01

- Combine Like Terms

Firstly, move all the terms involving the variable to one side of the equation. Subtract \(2x\) from both sides: \[ 2x - 5 - 2x = 7x - 2x \] which simplifies to \[ -5 = 5x \]
02

- Solve for x

Next, solve for \(x\) by dividing both sides of the equation by 5: \[ x = -\frac{5}{5} \] which simplifies to \[ x = -1 \]

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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 the terms that have the same variables raised to the same power. Only coefficients of like terms can be added or subtracted. In the equation \[ 2x - 5 = 7x \], first \[ 2x \] and \[ 7x \] are the like terms. To combine them, move all terms involving the variable to one side. Subtract \[ 2x \] from both sides:
\[ 2x - 5 - 2x = 7x - 2x \]
This leaves you with: \[ -5 = 5x \]. Here, \[ 2x \] on the left cancels out, simplifying the equation.
Isolating Variables
Isolating the variable means getting the variable by itself on one side of the equation. This helps you to find its value. From our simplified equation, \[ -5 = 5x \], the variable \(x\) is already isolated on one side. To solve for \(x\), divide both sides by the coefficient of \(x\), which is 5:
\( x = \frac{-5}{5} \)
This simplifies to \( x = -1 \). The goal is to perform operations that simplify the equation until the variable is isolated.
Simplifying Equations
Simplifying equations makes them easier to solve. Simplification involves combining like terms and reducing the equation to its simplest form. In the example \[ 2x - 5 = 7x \], after combining like terms, we simplified it to \[ -5 = 5x \]. Simplify further by dividing by the coefficient to isolate \(x\). Each step minimizes complexity and brings the equation closer to a simple, solvable form.
Basic Algebra
Basic algebra forms the foundation for solving linear equations. It involves manipulating algebraic expressions and applying arithmetic operations in a logical sequence. For the equation \[ 2x - 5 = 7x \], basic algebraic operations included:
  • Combining like terms
  • Isolating the variable
  • Simplification
These steps show how basic algebraic rules are applied to find the solution \(x = -1\). Understanding these basic concepts allows tackling more complex equations with confidence.

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