Linear Equations Cheat Sheet

Linear equations represent straight lines, and understanding their properties like slope, intercepts, and intersection points is key to solving problems in various mathematical and real-world contexts.

Core Principles

A linear equation has variables raised only to the first power or constants, forming a straight line when graphed.
The slope (m) of a line indicates its steepness and direction: positive for increasing y with x, negative for decreasing y with x. A larger absolute slope means a steeper line.
Lines are parallel if their slopes are equal (or both undefined), and perpendicular if the product of their slopes is -1.
The intersection of two lines is the point (x, y) that satisfies both equations simultaneously, solvable through substitution or elimination.

Action Steps

To find the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$
To determine if lines are parallel, check if their slopes are equal.
To determine if lines are perpendicular, check if the product of their slopes is -1.
To find the intersection of two lines, set their equations equal to each other (if in y= form) or use substitution/elimination to solve the system of equations.

Formulas

General Linear Equation: $ax + by = c$
Slope Formula: $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-Intercept Form: $y = mx + c$
Factored Form: $y = m(x - b)$
Point-Slope Form: $y - y_1 = m(x - x_1)$

Key Terms

Linear Equation: An equation where each term has only one variable to the first power or is a constant.
Slope (m): The ratio of the vertical rise to the horizontal run of a line, indicating its steepness and direction.
y-intercept (c): The y-coordinate of the point where a line crosses the y-axis.
x-intercept (b): The x-coordinate of the point where a line crosses the x-axis.
Parallel Lines: Lines with equal slopes (or both undefined), meaning they never intersect.
Perpendicular Lines: Lines whose slopes multiply to -1, meaning they intersect at a 90-degree angle.
Simultaneous Equations: A set of equations that are solved together to find a common solution, often used to find the intersection point of lines.

Real World Examples

Total sales for a sportswear store were $150,000 in year 3 and $250,000 in year 5.: A linear model can represent this data by treating (year, sales) as ordered pairs. The slope would be $(\$250,000 - \$150,000) / (5 - 3) = \$50,000$ per year. Using point-slope form, an equation can be found to estimate sales for other years.