HOW TO FIND A MISSING COORDINATE IN AN ORDERED PAIR USING SLOPE

Example 1 :

The line joining the points A(-2 , 3) and B(a , 5) is parallel to the line joining the points C(0 , 5) and D(-2 , 1). Find the value of a.

Solution :

If two lines are parallel, then slopes will be equal. Here the line joining the points AB and CD are parallel.

So, slope of AB  =  slope of CD

(1)  =  (2)

2/(a + 2)  =  2

2  =  2(a + 2)

2  =  2a + 4

2a  =  2 - 4

a  =  -1

So, the missing coordinate is -1.

Example 2 :

The line joining the points A(0, 5) and B(4, 2) is perpendicular to the line joining the points C(-1, -2) and D(5, b). Find the value of b.

Solution :

If two lines are perpendicular, then the product of slopes will be equal to -1.

slope of AB ⋅ slope of CD  =  -1

(-3/4) ⋅ [(b + 2)/6]  =  -1

-3(b + 2)/24  =  -1

b + 2  =  8

b  =  8 - 2  =  6

So, the missing coordinate is 6.

Example 3 :

Line m passes through the points (k, 7) and (3, k - 4) in the xy-plane. If the slope of line m is 3, what is the value of k ?

Solution :

Given that, slope = 3

[(k - 4) - 7] / (3 - k) = 3

(k - 4 - 7) / (3 - k) = 3

(k - 11) / (3 - k) = 3

k - 11 = 3(3 - k)

k - 11 = 9 - 3k 

k + 3k = 9 + 11

4k = 20

k = 20/4

k = 5

So, the value of k is 5.

Example 4 :

In the xy-plane, a line with slope of 1/3 passes through the points (-1, 1) and (5, n). What is the value of n ?

a) 9    b)   4   c)  3   d) 7/3

Solution :

Slope of the line = (n - 1)/(5 + 1)

Given that, slope = 1/3

1/3 = (n - 1)/6

6 = 3(n - 1)

6 = 3n - 3

3n = 6 + 3

3n = 9

n = 9/3

n = 3

Example 5 :

Line p is shown in the xy-plane, and the point with coordinates (5, r) lies on the line p. What is the value of r ?

a) -5/3    b)  -11/6    c)  -11/7    d) -12/7

Solution :

Equation of the line shown :

y = mx + b

y-intercept (b) = -6

y = m x - 6

The line passes through the point (7, 0), we get

0 = m(7) - 6

7m = 6

m = 6/7

y = (6/7)x - 6

Since the line passes through the point (5, r), we get

r = (6/7)5 - 6

r = 30/7 - 6

r = (30 - 42)/7

= -12/7

So, option d is correct.

Example 6 :

6x + y = 17

hx - 9y = -8

In the given pair of equations, h is  a constant. The graphs of the equations in the xy-plane are perpendicular lines. What is the value of h ?

Solution :

Slope of the line 6x + y = 7

y = -6x + 7

m₁ = -6

9y = hx + 8

y = (h/9)x + (8/9)

m₂ = h/9

m₁ x m₂ = -1

-6(h/9) = -1

2h/3 = 1

h = 3/2

So, the value of h is 3/2.

Example 7 :

The linear function f is defined by f(x) = ax + b where a  and b are constants. If the graph of y = f(x) in the xy-plane has a slope of 6 and passes through the point (-5, -14). What is the value of b ?

Solution :

f(x) = ax + b

Slope (a) = 6

Since the line passes through the point (-5, -14), we get

-14 = 6(-5) + b

-14 = -30 + b

-14 + 30 = b

b = 16

So, the value of b is 16.

Example 8 :

A line in the xy-plane, the line with equation ax - (1/3)y = 8, where a is a constant, passes through the point (2, 6). What is the x-coordinate of the x-intercept of the line ?

Solution :

ax - (1/3)y = 8

The line passes through the point (2, 6), we get

a(2) - (1/3) 6 = 8

2a - 2 = 8

2a = 8 + 2

2a = 10

a = 10/2

a = 5

Applying a = 5, we get

5x - (1/3)y = 8

To find x-intercept, put y = 0

5x - (1/3) (0) = 8

5x = 8

x = 8/5

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