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The Slope of a Line

Secondary 3-4

The slope (rate of change) of a segment or a line, generally denoted by the variable m, corresponds to the value of its incline with respect to the x-axis.

The slope of a line corresponds to the ratio of the difference of the y-coordinates and the difference of the x-coordinates of two points on the line.

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When the two points A(x1,y1) and B(x2,y2)  are given, the slope can be calculated using the following formula.

slope=m=ΔyΔx=y2y1x2x1

Calculate the slope of the following segment.

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slope=y2y1x2x1 

slope=421(4)

slope=25

The rate of change (slope) is therefore 2/5. This means that every time 5 units are moved on the positive x-axis, 2 units are moved up on the y-axis.

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Four different inclines can be found depending on the type of slope that is observed.

  • An increasing line has a positive slope.

  • A decreasing line has a negative slope.

  • A horizontal line has a slope of zero.

  • A vertical line has an undefined slope.

Increasing Line = Positive Slope

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Decreasing Line = Negative Slope

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Horizontal Line = Zero Slope

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Vertical Line = Undefined Slope

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Important!

With a horizontal segment, the slope is 0, because the numerator is equal to zero ( y2y1=0 ).

With a vertical segment, the slope is undefined, because the denominator in the slope calculation is zero ( x2x1=0 ). The result of a division by 0 is undefined.

It is possible to determine the slope of a line from the parameters of the equation when the latter is given.

 

General form

Ax+By+C=0

Functional form

y=mx+b

Symmetric form

xa+yb=1

Slope

AB

m

ba

Find out more!

In a relation between two variables represented by a linear function, the slope is defined as the rate of change.

Exercises

Exercise

The Slope of a Straight Line

Mathematics Secondary3-4