Linear Relationships Cheat Sheet

Mastering linear relationships involves understanding their equations, graphing techniques, and properties like gradient and intercepts to solve algebraic and real-world problems.

Core Principles

Equation of a line (y = mx + c): 'm' represents the gradient (steepness) and 'c' represents the y-intercept (where the line crosses the y-axis).
X-intercept: The point where a line crosses the x-axis (y=0).
Y-intercept: The point where a line crosses the y-axis (x=0).
Parallel lines: Have the same gradient (m1 = m2).
Perpendicular lines: The product of their gradients is -1 (m1 * m2 = -1).

Action Steps

To determine if a point lies on a line, substitute the point's coordinates (x, y) into the line's equation and check if the equation holds true.
To sketch a line using intercepts, find the x-intercept (set y=0) and the y-intercept (set x=0), then plot these two points and draw a line through them.
To find the gradient of a line given two points (x1, y1) and (x2, y2), use the formula m = (y2 - y1) / (x2 - x1).
To rearrange an equation into y = mx + c form, isolate 'y' on one side of the equation.
To solve linear equations, isolate the variable (e.g., x) by performing inverse operations on both sides of the equation.
To solve linear inequalities, follow the same steps as solving linear equations, but remember to reverse the inequality sign if you multiply or divide by a negative number.

Key Terms

Gradient: A measure of the steepness of a line, calculated as the ratio of the vertical change to the horizontal change between any two points on the line.
Y-intercept: The y-coordinate of the point where a line crosses the y-axis.
X-intercept: The x-coordinate of the point where a line crosses the x-axis.
Parallel Lines: Lines that have the same gradient and never intersect.
Perpendicular Lines: Lines that intersect at a right (90-degree) angle; their gradients are negative reciprocals of each other.

Real World Examples

A theme park charges a $12 entry fee plus $4 per ride.: This can be represented by the linear equation C = 4n + 12, where C is the total cost and n is the number of rides. This allows calculation of total cost for any number of rides or the number of rides affordable within a budget.