6.3 The Slope Formula

In the previous lesson, we found the slope of a line using the formula . In this lesson, we will explore a formula that can be used to help us find the slope of a line when we are given the coordinates of two points on the line.

As covered in the previous lesson, to find the slope of a line, we can identify the rise and run, and then input them into the slope ratio as shown below.

The rise is 3. The run is 5.

Table 6.3.2

Steps	Solution
Label (1, 2) as point 1. Label (4, 5) as point 2. It does not matter which point you assign as point 1 and which point you assign as point 2.
Use the slope formula.
Substitute the values into the formula.
Simplify the numerator and the denominator.
If possible, simplify.

Table 6.3.3

Steps	Solution
Label (5, 9) as point 1. Label (1, n) as point 2.
Substitute the values into the formula.
Algebraically solve the equation.

Practice 2

Find the unknown value in each question.

a) (3, 7) and (9, n),

b) (n, 5) and (4, 1),

Homework 

1. Use the slope formula to find the slope of the line that passes through each pair of points.

   a) (1, 5) and (5, 9)

   b) (–3, 4) and (2, –1)

   c) (1, 4), (3, 9)

   d) (2, 5), (4, 0)

   e) (–3, 3), (4, –5)

   f) (4, –5), (1, –2)

   g) (0, –27) and (82, –3)

   h) (–4, –10) and (–35, –10)

   i) (135, 214) and (96, 172)

   j) (–423, 310) and (561, –465)

2. Find the unknown value in each question.

   a) (2, 3) and (5, n),

   b) (–2, n) and (2, 6),

   c) (n, 3) and (3, –1),

   d) (1, –4) and (n, –2),

   e) (0, 5) and (n, 1),

   f) (–1, n) and (5, –6),

   g) (1, 2) and (n, 6),

   h) (–3, n) and (2, 7),

3. Two adventurous hikers, Amelia and Ada, are racing to the top of two different hills in the region: Eagle Peak and Thunder Ridge. Both hills are known for their challenging slopes, but one is steeper than the other. Eagle Peak starts at the point (1, 3) and reaches the summit at the point (4, 9). Thunder Ridge begins at the point (2, 4) and peaks at the point (6, 10). Amelia would prefer to climb the steeper hill. Which one should she climb?

4. A ski resort is building a new slope for skiers. The base of the slope is at the point (0, 0) and the top is at the point (10, 30). What is the slope of the ski hill?

5. A bridge rises from the point (3, 4) to the point (7, 12). What is the slope of the bridge?

6. A subway tunnel begins at the point (–3, 10) and ends at the point (5, –8). What is the slope of the tunnel?

7. A person climbs down a ladder starting at the point (3, 6) and ends at the point (7, –4). What is the slope of the ladder?

8. Two hikers are documenting the incline of a mountain trail. The first hiker starts at the point (2, 5) and reaches the summit at the point (8, 17). What is the slope of the mountain trail?

Answers

1.

a) 1	b) –1	c)	d)	e)
f)	g)	h) 0	i)	j)

3. Eagle Peak slope = = ; Thunder Ridge = = . Amelia should climb Eagle Peak because it is steeper.

4. 3

5. 2

6.

7.

8. 2