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Chapter 2: Problem 68

\(8 x+4-8=7 x-1\)

Short Answer

Expert verified
The solution is \(x = 3\).

Step by step solution

01

- Simplify Both Sides

First, simplify both sides of the equation. On the left side: \[ 8x + 4 - 8 = 8x - 4 \] The equation simplifies to: \[ 8x - 4 \] On the right side: \[ 7x - 1 \]
02

- Move Variables to One Side

Subtract \(7x\) from both sides to get all the \(x\) terms on one side: \[ 8x - 7x - 4 = 7x - 7x - 1 \] This simplifies to: \[ x - 4 = -1 \]
03

- Isolate the Variable

Add 4 to both sides to isolate \(x\): \[ x - 4 + 4 = -1 + 4 \] This simplifies to: \[ x = 3 \]

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Key Concepts

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

Simplification
When dealing with linear equations, the first step is to simplify both sides as much as possible. Simplification makes the equation easier to solve. Start by combining like terms and removing any constants that you can.

For example, in the given exercise, the equation is: 8x + 4 - 8 = 7x -1
First, combine the constants on the left side: 8x + 4 - 8 simplifies to 8x - 4.
There's no need for simplification on the right side: 7x - 1.
The simplified equation is now: 8x - 4 = 7x - 1.

Now that the equation is simplified, you can move to the next step.
Variable Isolation
To solve the equation, you need to isolate the variable, which means getting the variable ( x ) by itself on one side of the equation.

The current equation is: 8x - 4 = 7x - 1
First, move the variable terms to one side by subtracting 7x from both sides: 8x - 7x - 4 = 7x - 7x - 1,
which simplifies to: x - 4 = -1.
Now, the equation has all the x terms on one side.

To isolate the variable completely, we handle the constants next.
Equation Balance
Balancing an equation is crucial because it keeps both sides of the equation equal to each other. This can be done by performing the same operation on both sides of the equation.

In the simplified equation:
x - 4 = -1
Add 4 to both sides to keep the equation balanced:
x - 4 + 4 = -1 + 4.
This simplifies to:
x = 3.
Always remember that whatever operation you perform on one side of an equation, you must also do to the other side to maintain equality.

Now the equation is solved, and the value of x is determined.

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