Linear Equations and Slope Fundamentals

Study Notes

A linear equation is an equation that can be written in the form y = mx + b, where m and b are constants, and x and y are variables.
This represents a straight line on a graph.
'm' is the slope of the line, which represents the rate of change of y with respect to x.
'b' is the y-intercept, which is the point where the line crosses the y-axis.
Linear equations can also be written in standard form: Ax + By = C, where A, B, and C are constants.

The slope of a line measures its steepness.
It is calculated as the change in y divided by the change in x between any two points on the line.
Formula: m = (y₂ - y₁) / (x₂ - x₁)
Positive slope indicates an upward trend.
Negative slope indicates a downward trend.
A slope of zero indicates a horizontal line.
An undefined slope indicates a vertical line.

Given two points: Find the slope using the formula. Then, substitute one of the points and the slope into the point-slope form (y - y₁ = m(x - x₁)) to solve for the equation.
Given the slope and a point: Use the point-slope form directly.
Given the y-intercept and the slope: Use the slope-intercept form (y = mx + b) directly.

Solving for a variable in a linear equation means finding the value of that variable that makes the equation true.
This is done using algebraic operations to isolate the variable.
The goal is to have the variable on one side of the equal sign and a numerical value on the other side.
Simplify both sides of the equation.
Isolate the variable term.
Perform inverse operations (addition, subtraction, multiplication, division) to isolate the variable.
Check the solution by substituting the value back into the original equation to verify it is correct.

Parallel lines: Two lines are parallel if they have the same slope but different y-intercepts.
Perpendicular lines: Two lines are perpendicular if the product of their slopes is -1. For example, if one line has a slope of 2, the perpendicular line will have a slope of -1/2.
Horizontal lines: Horizontal lines have a slope of zero (m = 0). Their equations are in the form y = b, where 'b' is the y-intercept.
Vertical lines: Vertical lines have an undefined slope. Their equations are in the form x = a, where 'a' is the x-intercept.

Modeling real-world situations: Linear equations can be used to model relationships between variables in various fields such as physics, economics, and engineering.
Calculating future values: Linear equations can be used to predict future values based on known data and trends.
Finding solutions to problems: Linear equations are crucial in solving real-world problems requiring analytical methods, such as calculating costs, distances, or areas.
Describing trends: Linear equations can graphically represent trends, making it easier to predict future outcomes.