The Equation of a Straight Line Is Usually Written This Way
Equation of a Straight Line
The equation of a straight line is usually written this way:
y = mx + c
What does it stand for?
/Slope (or Gradient) / Y Intercept
y = how far up
x = how far along
m = Slope or Gradient (how steep the line is)
c = the Y Intercept (where the line crosses the Y axis)
How do you find "m" and "c"?
- c is easy: just see where the line crosses the Y axis.
- m (the Slope) needs some calculation:
m = / Change in Y
Change in X
/
Knowing this we can work out the equation of a straight line:
Example 1
m / = / 21
/ = / 2
b = 1 (where the line crosses the Y-Axis)
Therefore y = 2x + 1Example 2
c = 0 m = -3
This gives us y = –3x + 0
We do not need the zero!
Example 3: Vertical Line
What is the equation for a vertical line?
The slope is undefined ... and where does it cross the Y-Axis?
In fact, this is a special case, and you use a different equation, not "y=...", but instead you use "x=...".
Like this:
x = 1.5Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5
Rise and Run
Sometimes the words "rise" and "run" are used.- Rise is how far up
- Run is how far along
m / = / rise
run
/
How to find the rule – given two coordinates
Step 1: How to “find” the gradient
Step 2: How to “find” the y-intercept
Substitute a coordinate into the rule
Example:
a)Step 1: (2;6) (1 ;-8)
m=
m =
m= 14
Step 2: y = 14x + c
Subst (2; 6) into rule:
6 = 14(2) + c
6 = 28 + c
-22 = c
y = 14x -22
General form to find gradient and y-intercept
The coordinates of every point lying on a straight line will “fit” the equation of the line and every point not lying on the line will not fit the equation.
Lines parallel to axes
Lines parallel to x - axes have a zero gradient.
Lines parallel to y – axes :
Finding the rule from a table:
1.
x / 1 / 2 / 3 / 4 / 5 / 6y / 7 / 9 / 11 / 13 / 15 / 17
First difference:
y-intercept:
Rule:
2.
x / 0 / 1 / 2 / 3 / 4 / 5 / 6y / 10 / 8 / 6 / 4 / 2 / 0 / -2
First difference:
Y-intercept:
Rule:
Finding the rule from the graph:
Parallel lines:
Parallel lines have the same gradient
y= 4 x + 6Parallel lines
y= 4x -10
Perpendicular lines:
m 1 x m 2 = -1
y = -3x + 1
y = x + 5
-3 x = -1
- Find the equation of the straight line through the point ( 1,3) and perpendicular to the line y = 2x + 3
Distance: Length of the line joining point A( x1, y1) to point B, (x2,y2)
Find the length of the line joining the points A ( 3,-5) and B ( 6,4)