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Chapter 2: Problem 4

Solve each equation. See Examples 1 and \(2 .\) $$ 15 x-5=7+12 x $$

Short Answer

Expert verified
The solution is \(x = 4\).

Step by step solution

01

Isolate Variable Terms

To solve the equation \(15x - 5 = 7 + 12x\), we want all terms involving \(x\) on one side. Subtract \(12x\) from both sides to get \(3x - 5 = 7\).
02

Eliminate Constant Terms

Add \(5\) to both sides of the equation \(3x - 5 = 7\) to isolate the term with \(x\). This results in \(3x = 12\).
03

Solve for x

Divide both sides by \(3\) to solve for \(x\): \(x = \frac{12}{3} = 4\).

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Key Concepts

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

Algebra
Linear equations are one of the foundation blocks of algebra. In this topic, we primarily deal with expressions that involve both numbers and variables. These variables usually represent unknown values we aim to find. Algebra teaches us to manipulate and work with these expressions efficiently by applying a set of rules.

Here are some of the basic operations you can perform with linear equations:
  • Addition and subtraction: You can add or subtract the same value from both sides of the equation without changing the solution.
  • Multiplication and division: You can multiply or divide both sides of the equation by the same nonzero number.
  • Distributive property: This allows you to multiply a term over a sum or difference, such as transforming \(a(b + c)\) to \(ab + ac\).
Grasping these operations aids in solving for the variable and understanding more complex algebraic concepts later down the road.
Equation Solving
Equation solving is all about finding the value of the unknown variable that satisfies the equation. This process often involves a series of steps to systematically simplify the equation.

In solving linear equations, like the one given in the exercise, follow these steps:
  • Combine like terms: If similar terms are on the same side of the equation, combine them.
  • Move variable terms to one side: Choose a side to gather all terms with the variable, using addition or subtraction.
  • Isolate the variable: Adjust the equation to have the variable by itself, often by dividing or multiplying.
These steps systematically break down the equation, making it easier to pinpoint the variable's value. Following this structure helps avoid errors and maintains clarity throughout the solution process.
Variable Isolation
Variable isolation is the strategy of removing all other terms from one side of an equation to solve for the variable itself. The main objective is to get the variable alone on one side of the equal sign, providing a clear view of its value.

Here's a breakdown of the variable isolation process:
  • Eliminate variable terms: Start by moving all variable terms to one side, as seen in the example where we subtract \(12x\) from both sides.
  • Clear constant terms: Add or subtract constant numbers to further simplify, like adding \(5\) after isolating the variable on one side.
  • Solve for the variable: Use division or multiplication, as shown by dividing by \(3\) to render \(x = 4\).
This systematic method ensures that the solution is straightforward and eliminates confusion, making it one of the key skills in solving linear equations effectively.

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