Fundamental ideas in algebra and mathematics are linear equations and finctions. They involve interrelationships between variables that can be shown on a graph as straight lines. In many disciplines, including physics, economics, engineering, and analysis, it is crucial to comprehend linear equations and functions. 

Let's look at the main characteristics of linear equations and functions:-

1. Linear Equations:-

Algebraic equations with a variable whose highest power is one are said have a linear form. It has the following form : y=mx+b

Here, the dependent variable is denoted by y, the independent variable is denoted by x, the line's slope (or gradient) is denoted by m, and the y-intercept, or the point at which the line crosses the y-axis, is denoted by b.

2. Slope-intercept Form :-

A typical technique to represent linear equations is in the slope-intercept form. The formulas is y=mx+b. 

The y-intercept (b) denotes the value of y when x is zero, but the slope (m) denotes the rate of change or steepness of the line.

3. Graphing Linear Equations:-

A coordinate plane can be used to graph linear equations. You can see the relationship between the variables by plotting two points and drawing a straight line between them.

4. Slope :-

A linear equation's direction and steepness are determined by its slope. It is calculated as the ratio of any two places on the line's vertical change ( rise) to its horizontal change (run).

5. Parallel and Perpendicular lines :-

Perpendicular lines and parallel lines both have the same slope and never cross. Perpendicular lines connect at a 90-degree angle and have slopes that are the reciprocal of each other's negative values.

6. Intercepts :-

There are two types of intercepts : the x-intercepts and the y- intercept. The x-intercepts is where the line crosses the x-axis (y=0), and the y- intercepts is where the line crosses the y-axis (x=0).

7. Point-Slope Form :-

A linear equation's point-slope form is: y-y1=m(x-x1).

It indicates a line with a slope of (m) that passes through the specified point (x1,y1).

8. Linear Function :-

A linear function is a particular kind of mathematical function that comes after a linear equation. It has a graph that is in a straight line and a constant rate of change. Predictions and models of real-world occurences  can both be made using linear functions.

9. Systems of Linear Equations :-

Multiple linear equations involving the same variables make up a system of linear equations. The set of values, that concurrency fulfill each equation is the system's solution. Solving systems of linear equations can be done using graphing, substitution, or elimination techniques.

10. Applications :-

There are numerous practical uses for linear equations and functions. They can be used to estimate rates, forecast population increase, simulate motion and pressure, allocate resources more effectively, and examine business costs, among other things.

Building a solid foundation in mathematics and problem-solving requires an understanding of linear equations and functions. It offers a framework for deciphering and analyzing the connections between variables and producing data-driven predictions.