HOW TO CHECK IF THE POINT LIES ON THE LINE

A point lies on a line if its coordinates satisfy the equation of the line.

Example 1 :

Does (3, -2) lie on the line with equation 5x-2y  =  20 ?

Solution :

By applying the given point (3, -2) into the given equation, we get

5(3)-2(-2)  =  20

15 + 4  =  20

19  =  20

Since the given point doesn't satisfy the given equation. we say that the given point is not on the straight line.

Example 2 :

Does (3, 4) lie on the line with equation 3x-2y  =  1 ?

Solution :

By applying the given point in the equation of the line, we get

3(3)-2(4)  =  1

9-8  =  1

1  =  1

Since the given point satisfies the equation, the given point lies on the line.

Example 3 :

Find k, if 

(a)  (3, 4) lies on the line with equation 3x-2y  =  k

(b)  (-1, 3) lies on the line with equation 5x-2y  =  k

Solution :

(a)  Since the given (3, 4) lies on the given line, it will satisfy the equation.

x  =  3 and y  =  4

3(3)-2(4)  =  k

9-8  =  k

k  =  1

So, the value of k is 1.

(b)  Since the given (-1, 3) lies on the given line, it will satisfy the equation.

x  =  -1 and y  =  3

5(-1)-2(3)  =  k

-5-6  =  k

k  =  -11

So, the value of k is -11.

Example 4 :

Find a given that :

(a)  (a, 3) lies on the line with equation y  =  2x-1

(b)  (-2, a) lies on the line with equation y  =  1-3x

Solution :

Since the given point (a, 3) lies on the line y = 2x-1, the point will satisfy the equation.

Here x  =  a and y  =  -3

-3  =  2a-1

-3+1  =  2a

2a  =  -2

a  =  -1

So, the value of a is -1.

(b)  (-2, a) lies on the line with equation y  =  1-3x

Since the given point (-2, a) lies on the line y = 1-3x, the point will satisfy the equation.

Here x  =  -2 and y  =  a

a  =  1-3(a)

a  =  1-3a

a+3a  =  1

4a  =  1

a  =  1/4

So, the value of a is 1/4.

Example 5 :

The line through the points (ℎ, 3), (4, 1) intersects the line 7x − 9y − 19 = 0, at right angle. Find the value of ℎ.

Solution :

Slope of the line passes through two point (h, 3) and (4, 1)

Slope = (y₂ - y₁)/(x₂ - x₁)

= (1 - 3) / (4 - h)

= -2/(4 - h)

Slope of the line 7x - 9y - 19 = 0

9y = 7x - 19

y = (7/9) x - (19/9)

Slope = 7/9

Since the above lines are intersecting at right angle, the product of their slopes will be equal to -1.

[-2/(4 - h)] x (7/9) = -1

14/9(4 - h) = 1

14 = 9(4 - h)

14 = 36 - 9h

9h = 36 - 14

9h = 22

h = 22/9

Example 6 :

Which statement correctly describes the relationship between the x- and y-coordinates at any given point on the line y = 2x – 1?

Tick your answers.

a) The y-coordinate is 1 less than the x-coordinate.

b) The x-coordinate is 1 less than the y-coordinate.

c) The y-coordinate is 1 less than double the x-coordinate.

d) The x-coordinate is 1 less than double the y-coordinate

Solution :

y = 2x - 1

Comparing the given equation with y = mx + b

Slope = 2/1

slope = changes in y / changes in x

When x = 1

y = 2 - 1 ==> 1

When x = 2

y = 4 - 1 ==> 3

Writing the points (1, 1) (2, 3)

x - coordinate is 1 less than the y-coordinate.

Example 7 :

Describe the relationship between the x- and y-coordinates at any given point on the line

y = 5x + 7

Three points are shown.

A) (7, 28)     B) (–3, –22)    C) (8, 45)

a) Work out the equation of the straight line that passes through points A and B.

b) Does point C lie on the same straight line? Show workings to justify your answer.

Solution :

Equation of the line passes through the points A and B.

A) (7, 28)     B) (–3, –22)

Slope = (-22 - 28) / (-3 - 7)

= -50 / (-10)

= 5

Equation of the line 

y - y1 = m(x - x1)

y - 28 = 5(x - 7)

y - 28 = 5x - 35

y = 5x - 35 + 28

y = 5x - 7

To check if the point C lies on the same line, we apply the point C. If the point is going to satisfy the equation, then we decide the point C lies on the line otherwise it is not.

C(8, 45)

45 = 5(8) - 7

45 = 40 - 7

45 ≠ 33

They are not equal, then the point C does not lie on the line.

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