# Slope of the Line Equation Worksheet

Slope of the Line Equation Worksheet
• Page 1
1.
If A = (4, 3), B = (5, 4) and C = (8, 7), then which of the following is correct?
 a. Triangle ABC is equilateral triangle b. Triangle ABC is right triangle c. Triangle ABC is scalane triangle d. A, B, and C are collinear

#### Solution:

Three points A, B, C are collinear the slope of the line joining A, B = slope of line joining B, C.
[Definition.]

The points are A(4, 3), B(5, 4), C(8, 7)

Slope of line joining A, B = ΔyΔx = 4 - 35 - 4 = 1

Slope of line joining B, C = ΔyΔx = 7 - 48 - 5 = 1

Here slope of line joining A, B = slope of line joining B, C = 1

Hence, the points A, B, C are collinear.

2.
What is the slope of a line on which for every point the $x$ coordinate is same?
 a. Infinitely small b. Infinitely large c. Undefined

#### Solution:

Let L be the line on which for every point the x coordinate is same.

So, let A(x, y1), B(x, y2) be two points on line L.

Slope of line L = Slope of line joining A,B = m = ΔyΔx
[Use the slope formula.]

= y2 -y1x - x

= y2 -y10 = Undefined
[Division by 0 is not defined]

3.
Find the slope of the line passing through the points A (7, 9) and B (12, 12).
 a. $\frac{3}{5}$ b. $\frac{19}{21}$ c. 8 d. - $\frac{19}{21}$

#### Solution:

The slope of the line joining the two points A (7, 9) and B (12, 12) = m = ΔyΔx

= 12 - 912 - 7 = 3 / 5
[Use the slope formula.]

4.
If $a$$b$, then what is the slope of the line joining the points P(5$a$, 6$b$) and Q(6$b$, 5$a$)?
 a. Independent of $a$, $b$ b. Dependent of $a$ c. Dependent of $a$, $b$ d. Dependent of $b$

#### Solution:

The slope of the line joining the points P(5a, 6b) and Q(6b, 5a) = m = ΔyΔx

m = 5a - 6b6b - 5a
[Use the slope formula.]

= - (5a - 6b5a - 6b)

m = - 1

Slope of the line is "- 1" which is independent of both a, b.

5.
What is the slope of the line on which for every point the ordinate is same?
 a. Undefined b. Infinitely large c. Infinitely small

#### Solution:

Let L be the line on which for every point the ordinate is same.

So, let P(x1, y) and Q(x2, y) be the two points on line L.

Slope of line L = Slope of line joining P,Q = m = ΔyΔx

= y - yx2 -x1
[Use the slope formula.]

= 0x2 -x1 = 0

6.
Find the angle of inclination of a line whose slope is $\sqrt{3}$.
 a. $\frac{5\pi }{4}$ b. $\frac{\pi }{4}$ c. d. $\frac{2\pi }{3}$

#### Solution:

Let θ be the angle of inclination of the line with slope (- 3)

tan θ = - 3
[Use the definition of slope.]

tan θ = - 3 = tan(2π3)
[0 ≤ θ < π]

θ = 2π3

7.
If $a$ ≠ - 1, then what is the slope of the line joining the points (- 4$a$, 4) , ($a$2 + 4, $a$2)?
 a. b. c. d.

#### Solution:

The two points are P(- 4a, 4 ) and Q(a2 + 4, a2)

Slope of line joining P, Q = m = ΔyΔx = a2 - 4a2 + 4 - (- 4a)
[Use the slope formula.]

m = a2 - 4a2 + 4a +4

= (a - 2)(a + 2)(a + 2)2

= (a - 2)(a + 2)
[Cancel the common factor.]

Slope of line joining P, Q is m = (a - 2)(a + 2)

8.
If $t$ > 1and $m$ is the slope of the line joining the points ($\frac{2}{{t}^{2}}$ , $\frac{4}{t}$) and (2$t$2 , 4$t$), then which of the following is correct?
 a. $m$ =1 b. $m$ < 1 c. $m$ =2 d. $m$ > 1

#### Solution:

The points are P(2t2 , 4t)and Q (2t2 , 4t)

Slope of line joining P,Q = m = ΔyΔx = (4t-4t)(2t2-2t2)
[Use the slope formula.]

= 4(t-1t)2(t-1t)(t+1t)

So, m = 2t +1t
[Cancel the common factor.]

2t +1t < 1
[ For t >1, t + 1t > 2.]

So, m < 1.

9.
Find the values of $k$ such that the points ($k$ + 2, 2), (2$k$ + 2, 4) and (2$k$ + 3, 2$k$ + 1) are collinear.
 a. - $\frac{1}{2}$ , 2 b. - $\frac{1}{2}$ , -2 c. $\frac{1}{2}$ , -2 d. $\frac{1}{2}$ , 2

#### Solution:

The points are A(k + 2, 2), B(2k + 2, 4) and C(2k + 3, 2k + 1)

Slope of line joining A, B = slope of line joining B,C .
[The three points A, B, C are collinear.]

4 - 22k + 2 - k - 2 = 2k + 1 - 42k + 3 - 2k - 2
[Use the slope formula.]

2k = 2k - 31

2k2 - 3k - 2 = 0

(2k + 1)(k - 2) = 0
[Factor.]

k = - 12 , 2.

10.
What is the slope of the line 3$x$ + 4$y$ + 14 = 0?
 a. - $\frac{4}{3}$ b. $\frac{14}{3}$ c. $\frac{7}{2}$ d. - $\frac{3}{4}$

#### Solution:

Given line is 3x + 4y + 14 = 0

y = (- 34)x + (- 14 / 4)
[Solve for y.]

Slope of the line = m = - 3 / 4
[The above equation is in the slope intercept form y = mx + c. ]