DIFFERENNT FORMS OF EQUATION OF A STRAIGHT LINE

General form (or) Standard form :

ax + by + c = 0

Slope intercept form :

y = mx + b

Point slope form :

(y - y1) = m(x - x1)

Vertical line :

x = a

Horizontal line :

y = b

Example 1 :

A straight line with the gradient of -2 passes through the point (4, -1). Find the equation of the line in slope intercept form.

Solution :

Slope of the required line = -2

The required line is passing through the point (4, -1).

Equation of the line :

(y - y1) = m(x - x1)

(y - (-1)) = -2(x - 4)

y + 1 = -2(x - 4)

Distributing -2, we get

y + 1 = -2x + 8

Subtract 1 on both sides, we get

y = -2x + 8 - 1

y = -2x + 7

Example 2 :

A straight line passes through the points A(1, 0) and B(3, 6). Find the gradient of the line and find equation in standard form.

Solution :

Slope (m) = (y2 - y1)/(x2 - x1)

Slope of the line joining the points A(1, 0) and B(3, 6).

x1 = 1, x2 = 3, y1 = 0 and y2 = 6

m = (6 - 0)/(3 - 1)

m = 6/2

m = 3

Equation of the line in slope intercept form :

y = mx + b

y = 3x + b ----(1)

Applying the point (1, 0), we get

0 = 3(1) + b

b = -3

Applying the value of b in (1), we get

y = 3x - 3 (Slope intercept form)

Converting into standard form, we get

3x - y - 3 = 0

Example 3 :

4x + 3y = 6 is a straight line. Find the gradient and y-intercept of the line.

Solution :

4x + 3y = 6

Subtracting 4x on both sides.

3y = -4x + 6

Dividing by 3 on both sides.

y = (-4/3)x + (6/3)

y = (-4/3)x + 2

Slope (m) = -4/3 and y-intercept = 2.

Example 4 :

Write the equation of the line which is parallel to the y-axis and passes through (1, 4)

Solution :

Equation of the line parallel to y-axis is y = 4.

Example 5 :

Write the equation of the line which is parallel to the x-axis and passes through (3, -5) ?

Solution :

Equation of the line parallel to x-axis is y = 3.

Example 6 :

Find the slope and y-intercept of the lines given below.

(i) 4x + y = 9

(ii)  3x - 2y = 4

Solution :

Converting the given standard form to slope intercept form, we can find slope and y-intercepts.

(i) 4x + y = 9

Subtract 4x on both sides.

y = -4x + 9

Comparing it into y = mx + b, we get

Slope = -4 and y-intercept = 9

(ii)  3x - 2y = 4

3x = 2y + 4

Subtract 4 on both sides.

3x - 4 = 2y

Divide by 2 on both sides.

y = (3/2)x - (4/2)

y = (3/2)x - 2

Slope (m) = 3/2 and y-intercept = -2.

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