## How to find the slope of a line

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The notion of slope is something you find many times in **normal life**. Consider pushing a wagon down some ramp or climbing up stairs. The stairs and ramp both have a slope. It’s possible to describe that slope, or the item’s steepness by considering **vertical** as well as **horizontal** movement along them. When we speak, words such as **“gradual”** or **“steep”** are used for describing slope. Along a **gradual** slope, the majority of movement will be **horizontal.** Alongside a **steep** slope, **vertical** movement is greater.

### Defining Slope

The **mathematical definition** of slope is quite similar to our everyday one. With math, slope is utilized for describing the **steepness** and **direction** of the lines. Via merely looking at the graph of a line, it’s possible to learn things about its slope, particularly in **relation** to other lines graphed on the exact same coordinate plane. Think about the graphs of the 3 lines below:

Firstly, study green and red lines green and red. If one thinks of the lines as hills, they’d say the red line is **steeper** than the green line. the red line has a **greater slope** than the green line.

After that, see that green line and red line slant **upward** as one moves from left to right. You could say that these 2 lines have a **positive slope.** The yellow line slants **downward** from left to right. The yellow line has a **negative slope.** Utilizing 2 of the points on the line, it’s possible to find the line’s slope via discovering the **rise** and the **run.** The **vertical** change between 2 points is known as a **rise,** and the **horizontal** change is known as a **run.** The slope equals the rise divided by the run: \(slope \ = \ \frac{rise}{run}\)

### Discovering the Slope of a Line if Given Two Points

We have seen that it’s possible to discover a line’s slope on a graph via measuring its rise and the run. It is also possible to discover the slope of a straight line without a graph if one knows the **coordinates** of any two points on that line. Every point has a set of coordinates: an \(x\)-value and a \(y\)-value, written as an ordered pair \((x, y)\). The \(x\) value informs one **where** a point is **horizontally.** The \(y\) value informs one **where** the point is **vertically.**

Think about 2 points on a line, Point \(1\) and Point \(2\). Point \(1\) has coordinates \((x_{1}, y_{1})\) and Point \(2\) has coordinates \((x_{2}, y_{2})\).

Its **rise** is the **vertical distance** between the \(2\) points, which is also the difference between their \(y\)-coordinates. That makes the rise \(y_{2} \ − \ y_{1}\). The run in between these \(2\) points is the **difference** in the \(x\)-coordinates, or \(x_{2} \ − \ x_{1}\).

So, \(slope \ = \ \frac{rise}{run} \ \ \) or \( \ m \ = \ \frac{y_{2} \ − \ y_{1}}{x_{2} \ − \ x_{1}}\)

### Discovering the Slopes of Horizontal and Vertical Lines

Up to now, we’ve considered lines which run **“uphill”** or **“downhill.”** Their slopes might be either **steep** or **gradual,** however, they’re always going to be **negative** or **positive** numbers. However, there are 2 additional types of lines, **vertical** and **horizontal.** What’s the slope of a flat line or level ground? Or of a wall or a vertical line?

So, think about a **horizontal** line on a graph. It doesn’t matter which two points are picked on the line, they’ll forever have the **same y-coordinate**. The equation for this line is \(y \ = \ 2\). The equation additional can be written as \(y \ = \ (0)x \ + \ 2\).

Utilizing the form \(y \ = \ (0)x \ + \ 2\), one will see the slope is \(0\). One can additionally utilize the slope formula with 2 points on this horizontal line to determine the slope of this horizontal line. Utilizing \((1, 2)\) as Point \(1\) and \((2, 2)\) as Point \(2\), one gets:

\(m \ = \ \frac{y_{2} \ − \ y_{1}}{x_{2} \ − \ x_{1}} \ = \ \frac{2 \ − \ 2}{2 \ − \ 1}\)

The slope of this horizontal line is \(0\).

So, think about any horizontal line. It **doesn’t** which 2 points on the line are picked, they’ll **forever** have the same \(y\)-coordinate. Therefore, whenever one applies the slope formula, the **numerator** is always going to end up as \(0\). Zero divided by **any** non-zero number is \(0\), therefore, the slope of any horizontal line is forever \(0\).

What about **vertical** lines? In this case, it doesn’t matter what 2 points are picked, they’ll forever have the** same x-coordinate**. The equation for this line is \(x \ = \ 3\).

There’s no way this equation can be put into the slope-point form since the **coefficient** of \(y\) is \(0\) \((x \ = \ 0y \ + \ 3)\).

Therefore, what occurs whenever one uses the slope formula with 2 points on this vertical line to determine the slope? Utilizing \((3,2)\) as Point \(1\) and \((3,-1)\) as Point \(2\), one comes up with:

\(m \ = \ \frac{y_{2} \ − \ y_{1}}{x_{2} \ − \ x_{1}} \ = \ \frac{-1 \ − \ 2}{3 \ − \ 3}\)

However, **dividing** by zero **doesn’t** have a **meaning** for the set of real numbers. Due to that fact, it’s said the slope of this vertical line is **undefined.** That’s true for all vertical lines. they will **all** have an **undefined slope**.

### Exercises for Finding Slope

**1) **\(y = 4x - 4\)\( \ \Rightarrow \ \)

**2) ** \(2y = -6x - 3\)\( \ \Rightarrow \ \)

**3) **\(30x + 6 - 3y = 0\)\( \ \Rightarrow \ \)

**4) **\(9x + 3 - 3y = 0\)\( \ \Rightarrow \ \)

**5) **\(-10x + 5 - 2y = 0\)\( \ \Rightarrow \ \)

**6) **Find the slope of the line passing through \((4, 1),(0, -75) \)\( \ \Rightarrow \ \)

**7) **\(y = -3x - 5\)\( \ \Rightarrow \ \)

**8) **Find the slope of the line passing through \((2, 7),(5, 43) \)\( \ \Rightarrow \ \)

**9) **Find the slope of the line passing through \((4, 1),(3, 4) \)\( \ \Rightarrow \ \)

**10) **Find the slope of the line passing through \((0, 0),(-3, 30) \)\( \ \Rightarrow \ \)