The slope of a line calculates the “steepness” of a line. It is usually denoted by the letter m. So, the slope of a line is the change in Y divided by the change in X. As the change in Y is very high, the slope can range from zero to any number that we can think of. However, we usually have a maximum slope of positive or negative infinity. The change in x is much smaller than the change in y, which means that the change in x is much less than the change in y.
The slope of a line shows how slant the line is, how much the line rises vertically is compared with how much it runs horizontally. Being able to find the slope of a line, or using the slope to find points on the line, is an important skill used in economics, geoscience, accounting/finance, and other fields. The slope of a line is also defined as the ratio of rise over run.
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The x and y coordinates of the lines are used to calculate the slope of the lines. It is the ratio of the change in the y-axis to the change in the x-axis.
The formula to calculate slope is given as,
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Slope of a Line m = [y2 – y1 / x2 – x1] |
That is,
[frac{text{change in y Coordinates}}{text{change in x Coordinates}}]
Where m is the slope of the line. x1, x2 are the coordinates of x-axis and y1, y2 are the coordinates of y-axis.
The slope of the line can be a positive or negative value.
x and y are only used to identify the two points. They are not values or exponents, the points can be given any names.
[m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]
Slope formula when the general equation of a straight line is given
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If the general equation of a straight line is given as
ax + by + c = 0,
then, the formula for the slope of the line is
m = – coefficient of x / coefficient of y
m = -a/b
Slope Intercept Formula
Linear equations are “straight line” equations that have simple variable expressions with terms without exponents on them. You are dealing with a straight line equation, if you come across an equation with only x and y. To find the equation of a line and y-intercept in the steepness of the line, we use slope intercept formula.
Slope intercept formula is
The values used in formula are as follows:
Rules for Calculating the Slope of Line
Method 1:
Step 1: Find two points on the line.
Step 2: Count the rise. The number of units counts up or down to get from one point to the next. Record this number as your numerator.
Step 3: Count the run. The number of units counts left or right to get to the point. Record this number as your denominator.
Step 4: Simplify your fraction if possible.
Method 2:
To find out the slope of a line, we need only two points from that line, (x1, y1) and (x2, y2).
There are three steps for calculating the slope of a straight line.
Step1: Identify two points on the line.
Step2: Select one to be (x1,y1) and the other to be(x2,y2).
Step3: Use the slope of the line formula to calculate the slope.
Some of the important points to remember to find the slope of the line. They are as follows:
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The slope formula can give a positive or negative result.
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If the slope is a positive value, the line is in a rising state.
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If the slope is a negative value, the line is descending.
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Vertical lines have no slope.
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Horizontal lines have a zero slope.
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Parallel lines have equal slopes.
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Perpendicular lines have negative reciprocal slopes.
Solved Examples
Example 1: Find the slope of the line whose coordinates are (2,6) and (5,1).
Solution:
We have,
(x1, y1) = (2, 6)
(x2, y2) = (5, 1)
The slope of a line formula is [m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]
m = (1 − 6/ 5 − 2)
m = −5/3
m = − 1.666
Example 2: If the slope of a line passing through the points (4, x) and (2, -7) is 3, then what is the value of x?
Solution:
We have
Slope = m = 3
Points:
(x1, y1 ) = (4, x)
(x2, y2) = (2, -7)
We know that,
Slope [m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]
3 = (-7 – x)/(2 – 4)
3 = (-7 – x)/(-2)
or, -7 – x = 3(-2)
or, -7 – x = -6
so, x = -7 + 6 = -1
Therefore, the value of x = -1.