Linear Equations and Coordinate Geometry

Linear Equations in Two Variables

• A linear equation in two variables is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
• The graph of a linear equation in two variables is a straight line.
• The solution to a linear equation in two variables is an ordered pair (x, y) that satisfies the equation.
• Infinite solutions exist for linear equations.
• The slope of a line can be calculated using two points on the line.
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• Systems of linear equations can be solved graphically or algebraically.
• Solving graphically involves finding the intersection point of the lines.
• Solving algebraically involves methods like substitution or elimination.

Coordinate Geometry

• Coordinate geometry is a branch of mathematics that uses coordinate systems to study geometric figures.
• The coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular number lines, the x-axis and the y-axis.
• Each point on the coordinate plane is uniquely identified by an ordered pair of coordinates (x, y), where x represents the horizontal position and y represents the vertical position.
• The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane can be calculated using the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²).
• Midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
• The concept of a locus is important in coordinate geometry; it is the set of all points satisfying a given condition.
• Lines and curves can be described using equations in a coordinate system.

Lines and Angles

• Lines are one-dimensional figures that extend infinitely in both directions.
• Angles are formed by two rays sharing a common endpoint.
• Angles are measured in degrees or radians.
• Complementary angles are two angles whose sum is 90 degrees.
• Supplementary angles are two angles whose sum is 180 degrees.
• Vertical angles are a pair of opposite angles formed by intersecting lines.
• Transversal is a line that intersects two or more other lines.
• Angles formed by transversals intersecting parallel lines have specific relationships, including congruent and supplementary angles.
• Parallel lines have the same slope when plotted on a coordinate plane.

Polynomials

• A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
• Polynomials can have one or more variables.
• Examples of polynomials: 4x³ + 2x² - x + 7 , 5y²
• Polynomials are classified according to the degree and the number of terms.
• Degree of the polynomial is defined as the highest exponent of the variable present.
• Adding and subtracting polynomials involves combining like terms.
• Multiplying polynomials uses the distributive property.
• Dividing polynomials can be performed using long division or synthetic division methods.

Heron's Formula

• Heron's formula is a formula for calculating the area of a triangle when the lengths of all three sides are known.
• It's used for finding the area of a triangle when sides a, b, and c are provided.
• Heron's Formula requires first calculating the semiperimeter (s) which is (a + b + c) / 2.
• The area of the triangle is then calculated using the formula: Area = √(s(s-a)(s-b)(s-c)) , where 's' is the semiperimeter and a, b, and c are the lengths of the three sides (a, b, and c).