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# Run|Definition & Meaning

## Definition

The **difference** in the **vertical coordinates **of any **two points on a line** is called the **rise** between those two points if the difference (final point’s ordinate – initial point’s ordinate) is positive. Otherwise, if the difference is negative, it is called the fall. The term represents the numerator of the slope equation, where slope = rise $\div$ run, and rise or fall = y$_f$ – y$_i$.

Figure 1 illustrates the rise and run of a straight line.

The formula describing the **“slope”** of a straight line connecting any two locations is known as the rise over run formula. The term **“rise”** refers to the variation in the y-coordinates between two places. The term **“run”** refers to the variation in the x-coordinates of two identical** places**.

**Slope of a Line**

In daily life, you frequently come across the **concept of slope**. Consider descending a slope with a cart or ascending some steps. There is a slope on the steps as well as the ramp. By taking both **horizontal and vertical movement** along the ramp and steps into account, you may define the slope, or steepness, of those features. In conversation, you characterize slope with **“gradual”** or **“steep.”**

Most of the movement is horizontal down a gradually inclining slope. The vertical movement is larger when the **slope is steep. **The concept of **slope in mathematics** is pretty close to how we normally define it.

Simply put, the slope is both a literal and mathematical term used to indicate a line’s steepness and direction. A line’s slope can be inferred from its **graph alone**, especially in relation to other lines shown on the **same coordinate plane.**

As** x rises, y rises,** or alternatively, when **x falls, y falls **visually. This indicates that as we travel from left to right on the graph, the line rises, and the slope is positive. If, on the other hand, as **x rises, y falls,** or alternatively, when **x falls, y** **rises**, then as we travel from left to right on the graph, the line descends, and the **slope is negative.**

As **x rises, y remains constant,** or alternatively, **y remains constant while x falls** as we travel from left to right on the graph, the line does not move up or down, visually indicating that it is horizontal and a constant function then the **slope is zero. **If **y takes** on any value and x takes on only one value visually, this means the line is fully vertical, and the **slope is undefined**.

Figure 2 illustrates the positive, negative, and zero slopes of a straight line.

## Rules for Implementation of Rise Over Run Formula

The slope of a straight line, which can be **negative, positive, or zero**, can be calculated using the **rise-over-run** formula. Remember the following main points:

- If a straight line is
**heading up**and from**left to right**, it is a rising line with a positive slope. - If a straight line is
**falling**with a negative slope if it is moving**downhill**and from*left to right*. - The slope of a
**straight line**will be 0 if it is**horizontal**. - The slope is
**unclear**whether the straight line is vertical.

Figure 3 illustrates the **equation of a straight line** in which m indicates the **slope** or **gradient** while b indicates the **y-intercept**.

The slope formula is as follows if $U(x_1,y_1)$ and $V(x_2,y_2)$ are the two points on a straight line:

m = $\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$

As a result, using the formula above, we can quickly determine the **slope of a line** connecting two locations. How much the line rises **(along the y-axis)** from one point to the other over the run **(along the x-axis)** is another way to describe the slope of a line between two locations**.**

The **change in y **is the fraction’s numerator, whereas the change in x is the fraction’s denominator. These are the slope situations and what they indicate for the** rise and run**

**Rise and run**have the same sign when the slope is positive (both are positive or both are negative).**Rise and run**have opposite signs whether the slope is negative (one is positive, the other is negative).- In the case of a slope of zero,
**rise and run**can both be zero. - Run and rise are both 0 when the
**slope is indefinite**(infinite).

** **Some Examples of Run

### Example 1

a) Using the rise-over-run method, determine the slope of the line connecting points X(0, 2) and Y(1, 3).

b) Using the rise-over-run method, determine the slope of the line connecting points X(0, 8) and B(2, 4).

c) Using the rise-over-run method, determine the slope of the line connecting points X(0, 7) and B(5, 3).

### Solution

a) As we know, the equation of a straight line:

Y= mx + b

Using the slope formula:

Slope (m) = $\dfrac{y_{2}-y_{1}}{ x_{2}-x_{1}}$

m = $\dfrac{3-2}{1-0}$

m = 1

Point X passes through the line, so substituting the value of x, which is equal to zero, and y, which is two, in the equation of the line to evaluate b:

2 = 1(0) – b

b = -2

so substituting all the values in the equation of the line, we have:

y = x – 2

b) Equation of line:

Y= mx + b

Using the slope formula:

m = $\dfrac{4-8}{2-0}$

m = -2

Point X passes through the line, so substituting the value of x, which is equal to zero, and y, which is two in the equation of the line to evaluate b:

8 = -2(0) – b

b = -8

so substituting all the values in the equation of the line, we have:

y = -2x – 8

c) Equation of line:

Y= mx + b

Using the slope formula:

m = $\dfrac{3-7}{5-0}$

m = – 0.8

Point X passes through the line, so substituting the value of x, which is equal to zero, and y, which is two, into the equation of the line to evaluate b:

7 = -2(0) – b

b = -7

so substituting all the values in the equation of line, we have:

y = -0.8x – 7

**Example 2**

When Ali looked at the graph, she saw that the lift had been 30 units and the run had been 6 units. What value should a line’s slope have?

**Solution**

As given, the rise is 30 units, and the run is 6 units, and we know the formula for slope is:

Slope = Rise / Run

Slope = 30/6

Slope = 5

*All images were created using GeoGebra.*