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Linear equations are gradient intercept, point-gradient, standard and general form, Gradient, midpoint and distance formulas.
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݉ – gradient (slope) ܾ – y-intercept
Useful for interpreting the relationship between variables x and y, graphing, and identifying m and b. Also useful for generating the equation of the line when m and b are given (ie from a graph).
ݔሺଵ, ݕଵሻ – a point on the line ݉ – gradient (slope)
Useful for plugging in a point on the line (once the gradient is known) to generate the equation of the line. It may then be changed into another form.
ܣ is non-negative and ܣ, ܤ, ܥ are relatively prime integers (no common factors)
Useful for finding both the x- and y-intercepts of the line and using the intercepts to graph the line.
All points have y-coordinateܾ
The line is horizontal with gradient ݉ ൌ 0 and y-intercept ܾ.
All points have x-coordinateܽ
The line is vertical with an undefined gradient.
ݔሺଵ, ݕଵሻ & ݔሺଶ, ݕଶሻ – points on the line
The formula is “rise over run”, which is the change in y divided by the change in x.
It is recommended to read the gradient from left to right. If the line goes up from left to right, the gradient is positive. If the line goes down from left to right, the gradient is negative.
The gradients of parallel lines are equal. Parallel lines never intersect.
The gradients of perpendicular lines are negative reciprocals of each other. Perpendicular lines intersect at 90° (right) angles.
ݔሺଵ, ݕଵሻ & ݔሺଶ, ݕଶሻ – coordinates
The formula can be broken up into two parts, the x-part and y-part. Each part is the middle (or average) of the x and y coordinates of the given points.
ݔሺଵ, ݕଵሻ & ݔሺଶ, ݕଶሻ – coordinates
The formula is derived from the Pythagorean Theorem ܽ ଶ^ ܾ ଶ^ ܿൌ ଶ. The legs of a right triangle are given by the difference in x- and y-coordinates (inside the brackets). The distance is the hypotenuse of this right triangle. This is calculated by the square root of the sum of the squares of the legs (right- hand-side of formula).
Given an equation in the form ൌ ܥ ࢟ܤ ࢞ܣ 0 and a point ݔሺଵ, ݕଵሻ
The equation of the line must be in the above form and the perpendicular (shortest) distance from the point to the line can be calculated by plugging in all values.
ܣ is non-negative and ܣ, ܤ, ܥ are relatively prime integers (no common factors)
Similar to standard form except it is solved for 0 (all terms on left side). Not very useful for knowing anything about the line without formulas. Recommend converting to another form.