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Chapter 3: Problem 22

Solve each equation. $$-4 x+6=-5 x-9$$

Short Answer

Expert verified
The solution is \(x = -15\).

Step by step solution

01

Simplify the Equation

Start by getting all the terms that involve the variable \(x\) on one side and the constant terms on the other side. To do this, we add \(5x\) to both sides to move the \(x\)-terms to the left-hand side of the equation.\(-4x + 5x + 6 = -9\). This simplifies to \(x + 6 = -9\).
02

Isolate the Variable

Now, subtract \(6\) from both sides to solve for \(x\).\(x + 6 - 6 = -9 - 6\). This simplifies to \(x = -15\).

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Key Concepts

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

Algebra Fundamentals
Understanding algebra fundamentally begins with recognizing variables, constants, and operations used to solve equations. Algebra is the language of mathematics where letters represent numbers. In our equation, \(-4x + 6 = -5x - 9\), **x** is the variable. It represents the unknown we want to find.
**Constants** are the known numbers like 6 and -9, which don't change. The goal of solving the equation is to find out what value of **x** makes the equation true when both sides are equal. Knowing these essentials sets the stage for deeper topics like simplifying equations and isolating variables, which we'll explore next.
Simplifying Equations
Simplifying an equation is about making it easier to understand and solve. This process involves gathering similar terms and making the equation more straightforward. For instance, in the original step, \(-4x + 6 = -5x - 9\), we start by wanting all terms with \(x\) on one side. This action is pivotal to simplifying equations:
  • We added \(5x\) to both sides, transforming the left side into \(x + 6\).
  • Essentially, we are combining like terms \(-4x + 5x\) to simplify.
  • The equation becomes \(x + 6 = -9\).
Notice how combining terms makes the equation leaner and ready for the next step. Simplifying reduces potential confusion and gets you closer to the solution.
Isolating Variables
Once the equation is simplified, the aim is to isolate the variable on one side to find its value. For \(x + 6 = -9\), we need to eliminate the +6 from the left side to have **x** alone:
  • Subtracting 6 from both sides balances the equation, as the same operation must apply to each side.
  • This results in \(x = -9 - 6\).
  • Simplifying further gives us \(x = -15\).
Isolating the variable involves zeroing in on our target, **x**. It's all about making one side just the variable itself, which reveals its value clearly and effectively. By understanding this process, you enhance your problem-solving tactics for a range of algebraic equations.

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