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Chapter 2: Problem 54

\(9 x+5=8 x+4\)

Short Answer

Expert verified
x = -1

Step by step solution

01

- Isolate the variable

Subtract 8x from both sides of the equation to get all x terms on one side: \[9x + 5 - 8x = 8x + 4 - 8x\]This simplifies to: \[x + 5 = 4\]
02

- Solve for x

Subtract 5 from both sides to isolate x: \[x + 5 - 5 = 4 - 5\]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.

Isolate the Variable
When solving linear equations, the first crucial step is to isolate the variable. This means you want to get the variable (often represented as x) on one side of the equation by itself. In the given exercise, we start with the equation \(9x + 5 = 8x + 4\). To isolate x, we need to eliminate the terms involving x from one side of the equation.

We achieve this by subtracting 8x from both sides. This is important because it balances the equation, keeping it equal. The equation then becomes:

\(9x + 5 - 8x = 8x + 4 - 8x\)
Simplifying this step-by-step, we get:
\(x + 5 = 4\).

By isolating the variable, we set the stage for solving the equation in the next steps.
Simplify Equations
After isolating the variable, the next step is to simplify the equation. Simplifying makes the equation easier to solve by either combining like terms or performing basic arithmetic operations on both sides of the equation.
Starting from:
\( x + 5 = 4\)
We need to get x by itself. So, we move the constant term (+5) from the left side to the right side. This is done by subtracting 5 from both sides:

\(x + 5 - 5 = 4 - 5\)
Which simplifies to:
\(x = -1\).
All terms involving x are now consolidated, and all constants are moved to the other side, allowing us to fully isolate x.
Linear Equations
Understanding what a linear equation is can help you solve them more effectively. A linear equation is any equation that graphs as a straight line in the Cartesian plane. These equations often take the form of \(ax + b = c\), where a, b, and c are constants.

In the given exercise, our starting equation \(9x + 5 = 8x + 4\) is a linear equation.
Linear equations are characterized by having variables raised to the first power only and no products of variables.
Here are some key characteristics:
  • They graph to straight lines
  • They typically have only one solution
  • They can be solved using straightforward arithmetic operations

Understanding these properties can give you a good foundation for solving any linear equation you encounter.

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