Simultaneous Equations & Coordinate Geometry

Study Notes

Simultaneous Equations

Simultaneous equations involve two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously.
Methods for solving include:
Substitution: One equation is solved for one variable, and the resulting expression is substituted into the other equation.
Elimination: Variables are eliminated through addition or subtraction to create an equation with only one variable.
Graphically: The intersection point(s) of the graphs of the equations represent the solution(s).
Solutions can be:
A unique solution (the lines intersect at a single point).
No solution (the lines are parallel and do not intersect).
Infinitely many solutions (the lines coincide).

Coordinate Geometry

Coordinates of a point represent its position in a two-dimensional plane (x, y).
The Cartesian plane or coordinate plane is formed by two perpendicular number lines, the x-axis, and the y-axis, intersecting at the origin (0, 0).
Key concepts
Distance between two points: The distance formula can be used to find the distance between two points (x₁, y₁) and (x₂, y₂).
Midpoint of a line segment joining two points: The midpoint (x_m, y_m) of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by:
x_m = (x₁ + x₂)/2
y_m = (y₁ + y₂)/2
Slopes and Equations of Lines
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated as (y₂ - y₁)/(x₂ - x₁).
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
The point-slope form of a linear equation is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
Parallel and Perpendicular Lines
Parallel lines have the same slope.
Perpendicular lines have slopes that multiply to -1.
Finding equations from parallel or perpendicular lines

Sequences

Sequences are an ordered list of numbers, usually following a defined pattern or rule.
Types of Sequences
Arithmetic: The difference between consecutive terms is constant (common difference).
Geometric: The ratio between consecutive terms is constant (common ratio).
Arithmetic Sequences
General term (nth term): a_n = a₁ + (n-1)d, where a₁ is the first term, d is the common difference, and n is the position of the term.
Sum of an arithmetic sequence: S_n = n/2(a₁ + a_n)
Geometric Sequences
General term (nth term): a_n = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the position of the term.
Sum of a finite geometric sequence: S_n = a₁ (1-rⁿ) / (1-r).
Other types
Fibonacci sequence: Each term is the sum of the two preceding terms.
Recurrence relations: A rule that defines each term in a sequence based on previous terms.

Description

Explore the concepts of simultaneous equations and coordinate geometry. This quiz covers solving methods such as substitution and elimination, as well as understanding points in a Cartesian plane. Test your knowledge on the types of solutions and graphical representations.