Examples with solutions for Linear Function y=mx+b: Calculate the slope from two points

Exercise #1

The line passes through the points (2,3),(0,1) (-2,3),(0,1)

Video Solution

Step-by-Step Solution

To find the slope of the line passing through the points (2,3)(-2, 3) and (0,1)(0, 1), we use the formula for the slope mm between two points (x1,y1) (x_1, y_1) and (x2,y2) (x_2, y_2) :

m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the given points (x1,y1)=(2,3)(x_1, y_1) = (-2, 3) and (x2,y2)=(0,1)(x_2, y_2) = (0, 1), we have:

m=130(2) m = \frac{1 - 3}{0 - (-2)}

This simplifies to:

m=22 m = \frac{-2}{2}

So, the slope is:

m=1 m = -1

Thus, the slope of the line is 1-1, corresponding to choice 2.

Answer

m=1 m=-1

Exercise #2

The line passes through the points (5,10),(0,0) (-5,10),(0,0)

Video Solution

Step-by-Step Solution

The problem asks us to find the slope of the line passing through the points (5,10)(-5, 10) and (0,0)(0, 0). To solve this, we'll follow these steps:

  • Step 1: Note the coordinates of the two points. Let (x1,y1)=(5,10)(x_1, y_1) = (-5, 10) and (x2,y2)=(0,0)(x_2, y_2) = (0, 0).
  • Step 2: Apply the slope formula m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1} .
  • Step 3: Substitute the values from the points into the formula and simplify.

Now, let's substitute and compute the slope:

m=y2y1x2x1=0100(5)=105 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 10}{0 - (-5)} = \frac{-10}{5} .

Simplifying, we get m=2 m = -2 .

Therefore, the slope of the line is m=2 m = -2 .

Answer

m=2 m=-2

Exercise #3

The line passes through the points (5,7),(1,3) (5,7),(1,3)

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Identify the coordinates of the points.

  • Apply the slope formula.

  • Calculate the slope value.

Let's work through the steps:

We are given two points on a line: (5,7)(5,7) and (1,3)(1,3).

Step 1: Assign the coordinates: (x1,y1)=(5,7)(x_1, y_1) = (5, 7) and (x2,y2)=(1,3)(x_2, y_2) = (1, 3).

Step 2: Use the slope formula m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}.

Substitute the coordinates into the formula:
m=3715=44=1 m = \frac{3 - 7}{1 - 5} \\ = \frac{-4}{-4} \\ = 1

Therefore, the slope of the line passing through the points (5,7)(5,7) and (1,3)(1,3) is m=1\bm{m = 1}.

Thus, the correct answer is m=1\bm{m = 1}, corresponding to choice 1.

Answer

m=1 m=1

Exercise #4

The line passes through the points (3,7),(6,14) (3,7),(6,14)

Video Solution

Step-by-Step Solution

To solve this problem, we'll calculate the slope of the line passing through the points (3,7) (3, 7) and (6,14) (6, 14) . The formula for the slope m m of a line through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

  • m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}

First, we identify our points as follows:
Point 1: (x1,y1)=(3,7)(x_1, y_1) = (3, 7)
Point 2: (x2,y2)=(6,14)(x_2, y_2) = (6, 14)

Next, apply the formula:
x1=3y1=7x2=6y2=14 x_1 = 3 \\ y_1 = 7 \\ x_2 = 6 \\ y_2 = 14 \\
Substitute into the slope formula:
m=14763=73 m = \frac{14 - 7}{6 - 3} = \frac{7}{3}

Therefore, the slope of the line is m=73=213 m = \frac{7}{3} = 2\frac{1}{3} .

The correct choice from the given options is: m=213 m=2\frac{1}{3} .

Answer

m=213 m=2\frac{1}{3}

Exercise #5

The line passes through the points (2,2),(9,16) (2,2),(9,16)

Video Solution

Step-by-Step Solution

To solve this problem, we'll calculate the slope of the line passing through the points (2,2)(2, 2) and (9,16)(9, 16).

  • Step 1: Identify the coordinates
  • Step 2: Apply the slope formula
  • Step 3: Simplify the expression

Let's proceed:

Step 1: The coordinates given are (x1,y1)=(2,2)(x_1, y_1) = (2, 2) and (x2,y2)=(9,16)(x_2, y_2) = (9, 16).

Step 2: The slope m m of a line through two points is given by:

m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the coordinates into the formula, we have:

m=16292 m = \frac{16 - 2}{9 - 2}

Step 3: Simplify the expression:

m=147 m = \frac{14}{7}

m=2 m = 2

Therefore, the slope of the line is m=2\mathbf{m = 2}.

Answer

m=2 m=2

Exercise #6

The line passes through the points (2,4),(2,4) (-2,-4),(2,4)

Video Solution

Step-by-Step Solution

To find the slope of the line that passes through the points (2,4)(-2, -4) and (2,4)(2, 4), we use the slope formula:

  • The slope m m is calculated using m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1} .
  • Plugging in the given points, (x1,y1)=(2,4)(x_1, y_1) = (-2, -4) and (x2,y2)=(2,4)(x_2, y_2) = (2, 4), we have:

m=4(4)2(2) m = \frac{4 - (-4)}{2 - (-2)}

m=4+42+2 m = \frac{4 + 4}{2 + 2}

m=84 m = \frac{8}{4}

After simplifying, we find:

m=2 m = 2

Therefore, the slope of the line is m=2 m = 2 , corresponding to choice 3.

Answer

m=2 m=2

Exercise #7

The line passes through the points (3,6),(10,20) (3,6),(10,20)

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify and assign coordinates
  • Step 2: Apply the slope formula
  • Step 3: Simplify the calculations

Let's proceed with each step:

Step 1: Assign coordinates from the given points:
(x1,y1)=(3,6) (x_1, y_1) = (3, 6) and (x2,y2)=(10,20) (x_2, y_2) = (10, 20) .

Step 2: Apply the slope formula, which is:

m=y2y1x2x1=206103 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 6}{10 - 3} .

Step 3: Calculate the slope:

m=147=2 m = \frac{14}{7} = 2 .

Therefore, the slope of the line passing through the points (3,6) (3, 6) and (10,20) (10, 20) is m=2 m = 2 .

The correct choice from the given options is m=2 m = 2 .

Answer

m=2 m=2

Exercise #8

The line passes through the points (0,0),(5,5) (0,0),(5,-5)

Video Solution

Step-by-Step Solution

To find the slope of the line passing through the points (0,0)(0, 0) and (5,5)(5, -5), we will use the slope formula. Let's follow these steps:

  • Step 1: Identify the points as (x1,y1)=(0,0)(x_1, y_1) = (0, 0) and (x2,y2)=(5,5)(x_2, y_2) = (5, -5).
  • Step 2: Substitute these values into the slope formula m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1} .
  • Step 3: Perform the calculation:
    m=5050=55=1 m = \frac{-5 - 0}{5 - 0} = \frac{-5}{5} = -1

The calculation shows that the slope m m is 1-1.

Therefore, the solution to the problem is m=1 m = -1 .

Answer

m=1 m=-1

Exercise #9

The line passes through the points (6,19),(12,20) (6,19),(12,20)

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the coordinates of the given points.
  • Step 2: Apply the slope formula.
  • Step 3: Perform the subtraction and division required by the formula.

Now, let's work through each step:
Step 1: We have the points (x1,y1)=(6,19)(x_1, y_1) = (6, 19) and (x2,y2)=(12,20)(x_2, y_2) = (12, 20).
Step 2: The formula for the slope mm is m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1} .
Step 3: Substituting the values, we get m=2019126=16 m = \frac{20 - 19}{12 - 6} = \frac{1}{6} .

Therefore, the slope of the line that passes through the points is m=16 m = \frac{1}{6} .

Answer

m=16 m=\frac{1}{6}