The slope is a wide concept used in algebra. The slope is used to measure the direction and also the steepness when two points are given. The slope is very essential for the calculation of the equation of the line by using the slope-intercept form or point-slope form or intercept form.

There are various types of methods to calculate the equation of the line and in all the types to calculate the equation of line slope is very essential. The slope is also used in economics to measure the rate at which changes of the measurement take place.

The slope is usually used to measure the steepness and the direction of the graphs. The slope is very essential for the calculation of many algebraic problems. The slope of the line is stated as the ratio of the substance that x rises as y rises for some substance.

The slope tells how much substance of the y needs to rise as x also rises. The slope remains constant.

The slope moves from left to right in a graph showing the steepness of the line. When the slope is positive the line of the graph moves upward from left to right. And when the slope is negative the line of the graph moves downward from left to right

The slope is the ratio of the vertical change of a line (rise) to the horizontal change of a line (run).

Slope = the vertical change of a line / the horizontal change of a line

Slope = rise / run

The slope equation can be derived by a fundamental principle using distance formulas and the points of intersection. For the calculation of the slope using two points like (x_{1}, y_{1}) and (x_{2}, y_{2}). The horizontal line passes through (x_{1}, y_{1}) and the vertical line must pass through (x_{2}, y_{2}).

These lines must intersect at (x_{2}, y_{1}). The vertical line is the rise and the horizontal line is the run. Let’s apply the distance formula to find the run and rise using (x_{2}, y_{2}), (x_{2}, y_{1}), and (x_{1}, y_{1}), (x_{2}, y_{1}).

**For rise**

Distance formula = √ ((x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2})

Put (x_{2}, y_{2}), (x_{2}, y_{1}) for finding the rise.

Distance formula = √ ((x_{2} – x_{2})^{2} + (y_{2} – y_{1})^{2})

= √ ((0)^{2} + (y_{2} – y_{1})^{2})

= √ (y_{2} – y_{1})2

= (y_{2} – y_{1})

Hence, the rise is y_{2} – y_{1}

**For run**

Distance formula = √ ((x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2})

Put (x_{1}, y_{1}), (x_{2}, y_{1}) for finding the run.

Distance formula = √ ((x_{2} – x_{1})^{2} + (y_{1} – y_{1})^{2})

= √ ((x_{2} – x_{1})^{2} + (0)^{2})

= √ (x_{2} – x_{1})^{2}

= (x_{2} – x_{1})

Hence, the run is x_{2} – x_{1}.

So, we can write the slope equation as:

**m = (y _{2} – y_{1}) / (x_{2} – x_{1})**

The slope is generally calculated by using the formula when the points are given. The points are the change in the x-axis and the change in the y-axis.

Example 1

Find the slope of the line when the points are (8, -5) and (9, 4)?

Solution

Step 1:Identify the points from the given problem.

x_{1} = 8, x_{2} = 9, y_{1} = -5, y_{2} = 4

Step 2:Write the general formula of the slope.

m = (y_{2} – y_{1}) / (x_{2} – x_{1})

Step 3:Put the values in the above formula of the slope.

m = (4 – (-5)) / (9 – 8)

m = (4 + 5) / (1)

m = 9/1

m = 9

Example 2

Find the slope of the line when the points are (-12, -5) and (32, 34)?

Solution

Step 1: Identify the points from the given problem.

x_{1} = -12, x_{2} = 32, y_{1} = -5, y_{2} = 34

Step 2: Write the general formula of the slope.

m = (y_{2} – y_{1}) / (x_{2} – x_{1})

Step 3: Put the values in the above formula of the slope.

m = (34 – (-5)) / (32 – (-12))

m = (34 + 5) / (32 + 12)

m = 39/44

m = 0.8864

Slope intercept form is generally a way to find the equation of the line. To calculate the equation of the line with the help of the slope-intercept form, the slope is very essential in this regard. By name of the slope-intercept form, we can conclude that slope is very important for this purpose. The slope-intercept form gives the accurate and perfect equation of the line.

The slope-intercept form is generally an equation of a straight line having an equation of the form.

y = mx + b

Where x and y are variables, m is the slope, and b is the y-intercept form. There are many online tools present for the calculation of this process. A y=mx+b calculator is an online tool used for the calculations of the slope-intercept form along with the slope of the line, and point-slope forms.

Example 1

Find the slope-intercept form of the line when the points are (8, -5) and (9, 4)?

Solution

Step 1: Identify the points from the given problem.

x_{1} = 8, x_{2} = 9, y_{1} = -5, y_{2} = 4

Step 2: Write the general formula of the slope.

m = (y_{2} – y_{1}) / (x_{2} – x_{1})

Step 3: Put the values in the above formula of the slope.

m = (4 – (-5)) / (9 – 8)

m = (4 + 5) / (1)

m = 9/1

m = 9

Step 4: Now take the general formula of slope-intercept form.

y = mx + b

Step 5: Find the y-intercept form by pointing any pair of points in the slope intercept form formula along with the calculated slope.

y = 9x + b

4 = 9(4) + b

4 = 36 + b

b = 4 – 36

b = - 32

Step 6: Put the values in the formula.

y = mx + b

y = 9x – 32

This is the required slope-intercept form of the given points.

Example 2

Find the slope-intercept form of the line when the points are (3, -9) and (14, 24)?

Solution

Step 1: Identify the points from the given problem.

x_{1} = 3, x_{2} = 14, y_{1} = -9, y_{2} = 24

Step 2: Write the general formula of the slope.

m = (y_{2} – y_{1}) / (x_{2} – x_{1})

Step 3: Put the values in the above formula of the slope.

m = (24 – (-9)) / (14 – 3)

m = (24 + 9) / (11)

m = 33/11

m = 3

Step 4: Now take the general formula of slope-intercept form.

y = mx + b

Step 5: Find the y-intercept form by pointing any pair of points in the slope intercept form formula along with the calculated slope.

y = 3x + b

-9 = 3(3) + b

-9 = 9 + b

b = -9 – 9

b = - 18

Step 6: Put the values in the formula.

y = mx + b

y = 3x – 18

This is the required slope-intercept form of the given points.

The slope is very essential for the calculation of slope-intercept. Slope and slope-intercept forms can be calculated by using formulas. The slope is a general term used to measure the steepness of the line while the slope-intercept form is a way to calculate the equation of the line.

Derivation of slope of line formula. Mathematics Stack Exchange. Retrieved January 6, 2022, from https://math.stackexchange.com/ questions/ 720984/ derivation-of-slope-of-line-formula.

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