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Chapter 3: Problem 18

Solve the equation if possible. $$ 4 x+27=3 x $$

Short Answer

Expert verified
The solution to the equation is \(x = 27\).

Step by step solution

01

Combine like terms

Subtract \(3x\) on both sides of the equation to collect like terms together. This will give \(4x - 3x = 27\), which simplifies to \(x = 27\).
02

Solve the Simplified Equation

The equation after combining like terms is \(x = 27\), which is already solved for \(x\).

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Key Concepts

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

Solving Equations
The process of solving equations is like finding the missing piece of a puzzle. When you solve an equation, you aim to find the value of the variable that makes the equation true. This involves a variety of strategies and techniques for simplifying and rearranging terms.
In the given example, the equation is \(4x + 27 = 3x\). To solve this equation, we proceed step by step to isolate the variable \(x\) on one side. The ultimate goal is to determine what value of \(x\) will make both sides of the equation equal. Solving equations is a fundamental skill in algebra that helps us understand relationships between quantities and allows us to find unknown values.
Combining Like Terms
Combining like terms is a crucial technique in simplifying equations. Like terms are terms that have the same variable raised to the same power, for example: \(4x\) and \(3x\).
When combining like terms, you add or subtract coefficients while keeping the variable part the same. In our equation, we need to combine \(4x\) and \(3x\) by subtracting \(3x\) from both sides. This step simplifies the equation to \(4x - 3x = 27\), which further simplifies to \(x = 27\).
This technique helps to reduce complex equations into simpler forms, making it easier to determine the solution.
Basic Arithmetic
Basic arithmetic involves adding, subtracting, multiplying, and dividing numbers. These operations form the foundation of most mathematical manipulations. In our exercise, we used basic arithmetic when subtracting \(3x\) from both sides of the original equation \(4x + 27 = 3x\).
Though it may seem straightforward, mastering these operations is essential, as they are frequently used in solving more complex algebraic problems. Practicing basic arithmetic enhances one's ability to manipulate numbers quickly and accurately in various contexts.
Algebraic Manipulation
Algebraic manipulation entails the rearrangement and simplification of algebraic expressions to uncover solutions. This process involves various mathematical operations and follows logical rules, aimed at isolating the variable.
In the example equation of \(4x + 27 = 3x\), algebraic manipulation helps us move \(3x\) from the right to the left side, transforming the equation into \(4x - 3x = 27\) to eventually solve for \(x = 27\).
  • Isolate variables by performing inverse operations.
  • Maintain balance by applying operations equally on both sides of the equation.
These careful rearrangements allow for the clear solution of the equation, demonstrating how powerful algebraic manipulation can be.

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