Loading subject…
Addition principle
If you add (or subtract) the same quantity to both sides of an equation, the resulting equation is equivalent to the original.
Cartesian coordinate plane
A grid formed by two perpendicular number lines, the x-axis (horizontal) and y-axis (vertical), used to locate points with ordered pairs $(x,y)$.
Conjecture
A statement that appears to be true based on observed examples, but has not yet been proven.
Distance formula
For points $A(x_1,y_1)$ and $B(x_2,y_2)$, the distance between them is
$$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Equivalent Equations
Equations that have exactly the same solutions.
Generalization
A general statement or rule made on the basis of specific examples.
Gradient (slope)
For two distinct points $A(x_1,y_1)$ and $B(x_2,y_2)$ on a line, the gradient is
$$m=\frac{y_2-y_1}{x_2-x_1},\quad x_2\neq x_1$$
Infinitely many solutions
Infinitely many ordered pairs satisfy both equations. Graphically, the lines are coincident (the same line written in two forms).
Line of best fit
A straight line drawn through the middle of a scatter plot so that points are (roughly) evenly distributed above and below it, used to model and predict relationships.
Midpoint formula
For endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$, the midpoint is
$$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$
Multiplication principle
If you multiply (or divide) both sides of an equation by the same non-zero quantity, the resulting equation is equivalent to the original.
No solution
No ordered pair satisfies both equations. Graphically, the lines are parallel (same gradient, different intercepts).
Parallel lines
Two non-vertical lines are parallel if they have the same gradient.
Perpendicular lines
Two non-vertical, non-horizontal lines are perpendicular if their gradients multiply to $-1$. If one line has gradient $m\,(m\neq 0)$, then a perpendicular line has gradient $-\frac{1}{m}$.
Point-slope form
A line through $(x_1,y_1)$ with gradient $m$ can be written as $$y-y_1=m(x-x_1)$$
Representation
The manner in which information or a relationship is presented (for example, a table, graph, diagram, equation, or written rule).
Scatter diagram
A graph of paired (bivariate) data points plotted on an $x$-$y$ plane to investigate the relationship between two variables.
Slope-intercept form
The equation of a non-vertical line can be written as $$y=mx+c$$ where $m$ is the gradient and $c$ is the $y$-intercept.
Solution (of an equation)
A value of the variable that makes the equation true when substituted into the equation.
Standard form (linear)
A line can be written as $$ax+by+c=0$$ with constants $a,b,c$ (often integers, and sometimes with a required sign convention such as $a>0$).
System of linear equations
A set of two or more linear equations involving the same variables, solved together to find values that satisfy every equation in the system.
Unique solution
Exactly one ordered pair $(x,y)$ satisfies both equations. Graphically, the two lines intersect at one point.