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Equation of a straight line
The general equation of a straight line is represented by:
y = mx + c where m = y/x = gradient c = intercept on y axis
where x and y are variables that can have any value so the value of y depends on a given value of x. This can be shown by the simplest equation of a straight line y = x given below where x and y have the same values.
For a given equation m is always a number which is the gradient. It will not appear in the equation if its value is 1. C is the intercept which is the point the line crosses the vertical y axis. If c does not appear in the equation then it has a value of 0 and crosses the y axis at 0 so such a line will always pass through the origin.
y = x
Drawing a graph given the equation of a straight line
Example 1
Given the equation:
y = 2x + 2
The graph of this line can be drawn by generating values of y given a range of values of x.
The best way to do this is by drawing a table and choosing values that start from negative and end in positive values so x values from -2 to 2.
Note the values plotted from the table are (-2,-2), (-1,0), (0,2), (1,4), (2,6).
From the graph and the equation the intercept is 2 or sometimes expressed as a co-ordinate (0,2) since x is always 0 at the point where the line crosses the y axis.
The gradient is quite steep (2) and is positive (moves upwards from left to right).
Example 2
Exercise 1
1. Draw the graph of y = x + 3 using values of x between -2 and 2 constructing a table to generate y values.
2. Draw the graph of y = 2x - 1 using values of x between -2 and 2 constructing a table to generate y values.
3. Draw the graph of y = -½x + 2 using values of x between -2 and 2 constructing a table to generate y values.
How to work out the equation of a straight line given the graph
Example 1
After having worked out the gradient m = 1 and the intercept c = -2 we simply substitute these values into the general equation of the line which gives:
y = x - 2
Example 2
The gradient m = 3 and the intercept on the y axis is c = 2 so simply substitute these values into the general equation y = mx + c to give the actual equation of the line:
y = 3x + 2
Exercise 2
1. Determine the equation of the line of the following graphs:
(each square length equals one unit)
a)
b)
Working out the gradient given two points on a straight line
If two points on a straight line are given then the following formula can be used to work out the gradient:
so gradient m = 2
so gradient m = -3