GED Mathematical Reasoning: Slope Of A Line

Slope refers to the steepness of a line and is denoted by a lowercase “m.” Mathematically speaking, the measure of the slope of a line is the ratio of the vertical change (which we call the “rise”) to the horizontal change (which we call the “run”). In fact, it is popular to refer to slope as being the “rise over run.”

Slope m = \frac{rise}{run}

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Looking at a graph, a line that RISES or goes UPHILL – from left to right – has a POSITIVE slope, whereas a line that FALLS or goes DOWNHILL – from left to right – has a NEGATIVE slope.

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I’d like to mention two special cases when it comes to slope: the slopes of horizontal and vertical lines.

A horizontal line has a slope of zero.

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Why? Because a horizontal line has lots of horizontal change or “run,” but no rise. And in terms of the slope ratio, a zero in the numerator of the fraction yields a result of zero.

m = \frac{rise}{run} = \frac{0}{lots} = 0

A vertical line has a slope that is undefined.

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Why? Because a vertical line has lots of vertical change or “rise,” but no run. And in terms of the slope ratio, a zero in the bottom of the fraction yields an undefined result.

m = \frac{rise}{run} = \frac{lots}{0} =  undefined

 

Calculate the slope using two points on the line

Another way to calculate the slope of a line is numerically using the coordinates of two points on the line. In general, we call these points:

(x_{1},y_{1}) and (x_{2},y_{2})

The formula for calculating the slope of a line passing through the points (x_{1},y_{1}) and (x_{2},y_{2}), which is based on the “rise over run” concept, is:

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

 

Example 1

State the slope of the line shown.

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Right off the bat, we know that the slope of this line will be negative since, from left to right, this line is going downhill. To calculate the numerical slope value, we will rely on the idea that slope is the ratio of “rise over run.”

To determine the rise and run we must first identify two places or points where the line goes through the cross-hairs of the grid. Then – from one point to the next – we’ll count the number of units of vertical change or “rise” and then the number of units of horizontal change or “run”.

Let’s focus on the two points (1, 2) and (2, 0).

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From one point to the next, the vertical change or “rise” is 2 units.

Speaking of the word “rise,” don’t let your mind be confused by the use of that word because our line ISN’T rising. We are just using the word “rise” to mean the vertical change. The rising or falling nature of the line is accounted for in the final answer – with a positive or negative sign.

Now, let’s go back to the graph and analyze the horizontal change or “run.”

The horizontal change or “run” between the two points is one unit.

Putting all this together, the ratio of “rise to run” is “two over one”, which reduces to just “2.” And since the line goes downhill, the final answer is a slope of NEGATIVE 2.

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m = \frac{rise}{run} = -\frac{2}{1} = -2

 

Example 2

Calculate the slope of the line passing through the points (1, -2) and (-3, -7)

 

To calculate the slope of the line that passes through these two points, we can use the slope formula. It may help to label the coordinates before applying the formula and be very careful with the positive and negative signs! In this case, (x_{1},y_{1}) is (1, -2) and (x_{2},y_{2}) is (-3, -7).

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When we substitute these values into the slope formula, we get:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(-7) - (-2)}{(-3) - (1)}

\frac{(-7) + 2}{(-3) - (1)} = \frac{-5}{-4} = \frac{5}{4}

So the final answer is m=\frac{5}{4}

It’s also important to know that the order of the points doesn’t matter. We can just as well let the point -3, -7 be the first point and the result would be the same.

coord13When we substitute these values into the slope formula, we get:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(-2) - (-7)}{(1) - (-3)}

\frac{(-2) + 7}{(1) + (3)} = \frac{5}{4}

Our final answer is m=\frac{5}{4}, which is the same result.

Since the slope is positive, we know that visually the line references in Example 2 RISES from left to right. And for every 5 units of vertical change or “rise”, there are 4 units of horizontal change or “run”. Put another way, the line goes UP 5 units for every 4 units it goes RIGHT.

 

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