Question

Find the slope of each line.
1)
(-1, 1), (-1, 4)

Answer 1

The slope is undefined

Explanation:

The equation of the line is x=-1 and this equation’s slope is undefined because it’s a vertical line.

Hope this helps and answers your question :)

Answer 2

The line presented has an undefined slope.

Explanation:

We are given two points of a line: (-1, 1) and (-1, 4).

Coordinate pairs in mathematics are labeled as (x₁, y₁) and (x₂, y₂).

• The x-coordinate is the point at which if a straight, vertical line were drawn from the x-axis, it would meet that line.
• The y-coordinate is the point at which if a straight, horizontal line were drawn from the y-axis, it would meet that line.

Therefore, we know that the first coordinate pair can be labeled as (x₁, y₁), so, we can assign these variables these "names" as shown below:

• x₁ = -1
• y₁ = 1

We also can use the same naming system to assign these values to the second coordinate pair, (-1, 4):

• x₂ = -1
• y₂ = 4

We also need to note the rules about slope. There are different instances in which a slope can either be defined or it cannot be defined.

Circumstance 1: As long as the slope is not equal to zero, there can be a

• positive slope,
  `(1)/(3)`
• negative slope,
  `-(5)/(6)`

Circumstance 2: If the slope is completely vertical (there is not a "run" associated with the line), there is an undefined slope. This is the slope of a vertical line. An example would be a vertical line (the slope is still zero).

Circumstance 3: If the line is a horizontal line (the line does not "rise" at all), then the slope of the line is zero.

Therefore, a slope can be positive, negative, zero, or undefined.

Now, we need to solve for the line we are given.

The slope of a line is determined from the slope-intercept form of an equation, which is represented as
`\text{y = mx + b}`.

The slope is equivalent to the variable m. In this equation, y and x are constant variables (they are always represented as y and x) and b is the y-intercept of the line.

We can do this by using the coordinates of the point and the slope formula given two coordinate points of a line:
`m=(y_2-y_1)/(x_2-x_1).`

Therefore, because we defined our values earlier, we can substitute these into the equation and solve for m.

Our values were:

• x₁ = -1
• y₁ = 1
• x₂ = -1
• y₂ = 4

Therefore, we can substitute these values above and solve the equation.

`\displaystyle{m = (4 - 1)/(-1 - -1)}\\\\m = (3)/(0)\\\\m = 0`

Therefore, we get a slope of zero, so we need to determine if this is a vertical line or a horizontal line. Therefore, we need to check to see if the x-coordinates are the same or if the y-coordinates are the same. We can easily check this.

x₁ = -1

x₂ = -1

y₁ = 1

y₂ = 4

If our y-coordinates are the same, the line is horizontal.

If our x-coordinates are the same, the line is vertical.

We see that our x-coordinates are the same, so we can determine that our line is a vertical line.

Therefore, finding that our slope is vertical, using our rules above, we can determine that our slope is undefined.