Gradients of a straight line and Gradient of a curve

In coordinate geometry, we make use of points in a plane.  A point consists of the x-coordinate called abscissa and the y-coordinate known as ordinate.  In locating a point on the x – y plane    x – coordinate is first written and then the y-coordinate.  For example, in a given point (a, b), the value of x is a and that of y is b.  Similarly, in a point (3, 5), the value of x is 3 and that of y is 5.  A linear graph gives a straight line graph from any given straight line equation which is in the general form y = mx + c or ax + by + c = 0

Example: Draw the graph of equation 4x + 2y = 5

1. Point of intersection of two linear equations

Two lines y = ax +b and y₂ = cx + d

Intercept when ax + b = cx + d

That is you solve the two equations simultaneously

1. Intersection of a line with the x or y axis

The point of intersection of a line with the x –axis can be obtained by putting y = o to find the corresponding value of x = a, say the required point of intersection gives (a, o).  Similarly, for the point of intersection of a line with the y-axis, put x = o to find the corresponding value of y.  If the corresponding value of y is b, the required point of intersection is (o, b)

Example: Find the point of intersection of the line 2x + 3y + 2 = 0 with the

1. x – axis (ii)        y – axis

Example 3: Find the point of intersection of the lines y = 3x + 2 and y = 2x + 5

Solution

y = 3x + 2         (1)

y = 2x + 5         (2)

At the point of intersection

3x + 2 = 2x + 5

3x – 2x = 5 -2

X = 3

Substitute 3 for x in equation (1), we obtain y = 3(3) + 2 = 11.

Hence, the point of intersection is (3, 11)

GRADIENT OF A STRAIGHT LINE

The Gradient of a straight line is defined as the ratio

Change in y in moving from one

Change in x point to another on the line.  The Gradient of a straight line is always constant.

TANGENT OF A SLOPE