An educational summary outlining the definitions and algebraic rules for working with linear equations in one variable.

The image you provided is a summary of the concepts related to linear equations in one variable. It explains what linear equations are, how to solve them, and introduces some fundamental properties and methods for solving such equations. Below, I will break down the key points and explain the solution process step by step.

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Key Concepts from the Image

1. Definition of Linear Equations:
- A linear equation is an equation of equality between two algebraic expressions involving one or more unknown quantities.
- For example: \( \frac{2x}{3} - 4 = \frac{x}{7} + \frac{7-x}{3} = 11 \).

2. Properties of Linear Equations:
- Any value of the variable that makes the equation true is called the solution or root of the equation.
- The process of moving terms from one side of the equation to the other with a sign change is called transposition.
- Without changing the equality, we can:
- Add the same quantity to both sides.
- Subtract the same quantity from both sides.
- Multiply both sides by the same non-zero quantity.
- Divide both sides by the same non-zero quantity.
- Cross multiplication: If \( \frac{ax+b}{cx+d} = \frac{p}{q} \), then \( q(ax+b) = p(cx+d) \).

3. Performing Mathematical Operations on Equations:
- When solving linear equations, any operation performed on one side of the equation must also be performed on the other side to maintain equality.
- Example:
- Given \( 4x + 3 = 3x + 7 \):
- Subtract 3 from both sides: \( 4x + 3 - 3 = 3x + 7 - 3 \).
- Simplify: \( 4x = 3x + 4 \).
- Subtract \( 3x \) from both sides: \( 4x - 3x = 3x + 4 - 3x \).
- Simplify: \( x = 4 \).

4. Methods for Solving Linear Equations:
- There are six main methods:
1. Trial and Error Method
2. Graphical Method
3. Elimination Method
4. Substitution Method
5. Transposition Method
6. Cross Multiplication Method

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Solution Process for Linear Equations

Let's solve a sample linear equation step by step using the principles outlined in the image.

#### Example Problem:
Solve the linear equation:
\[ \frac{2x}{3} - 4 = \frac{x}{7} + \frac{7-x}{3} \]

#### Step-by-Step Solution:

1. Eliminate Fractions:
- Find the least common multiple (LCM) of the denominators (3 and 7). The LCM is 21.
- Multiply every term by 21 to eliminate the fractions:
\[
21 \left( \frac{2x}{3} \right) - 21(4) = 21 \left( \frac{x}{7} \right) + 21 \left( \frac{7-x}{3} \right)
\]
- Simplify each term:
\[
7(2x) - 84 = 3(x) + 7(7-x)
\]
\[
14x - 84 = 3x + 49 - 7x
\]

2. Combine Like Terms:
- Combine the \( x \)-terms on the right-hand side:
\[
14x - 84 = 3x - 7x + 49
\]
\[
14x - 84 = -4x + 49
\]

3. Transposition:
- Move all \( x \)-terms to one side and constant terms to the other side:
\[
14x + 4x = 49 + 84
\]
\[
18x = 133
\]

4. Solve for \( x \):
- Divide both sides by 18:
\[
x = \frac{133}{18}
\]

#### Final Answer:
\[
\boxed{\frac{133}{18}}
\]

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Explanation of the Solution:

- We started by eliminating the fractions by multiplying through by the LCM of the denominators.
- After simplifying, we combined like terms and used transposition to isolate \( x \).
- Finally, we solved for \( x \) by dividing both sides by the coefficient of \( x \).

This method follows the principles outlined in the image, ensuring that the equality is maintained throughout the process.