Category: Graphing

What are Slope, Rise and Run?

To calculate slope, you will need to understand the concepts of rise and run.

Slope

The slope of a line is a measurement of how much the line is tilted.

The formula for slope is:

\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)

Rise

In the slope formula above, “rise” means the change in the y coordinates.

Run

On the other hand, “run” means the change in the x coordinates.

Slope – Free exercises

Instructions: Calculate the slope of the line by determining the rise and run of the following sets of coordinates.

1) (3,4) and (2,10)

2) (7,2) and (4,2)

3) (2,4) and (4, –6)

4) (4,2) and (–3,5)

5) (4,–3) and (–2,0)

Slope – Answers to the Exercises

Exercise 1: (3,4) and (2,10)

\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)

\(\frac{10 – 4}{2 – 3} = m\)

\(\frac{6}{–1} = m\)

–6 = m

Exercise 2: (7,2) and (4,2)

\(\frac{y2 – y1}{x2 – x1} = m\)

\(\frac{2 – 2}{4 – 7} = m\)

\(\frac{0}{–3} = 0 = m\)

So, the coordinates represent a horizontal line.

Exercise 3: (2,4) and (4, –6)

\(\frac{–6 – 4}{4 – 2} = m\)

\(\frac{–10}{2} = –5= m\)

Exercise 4: (4,2) and (–3,5)

\(\frac{5 – 2}{–3 – 4} = m\)

\(\frac{3}{–7} = –\frac{3}{7} = m\)

Exercise 5: (4,–3) and (–2,0)

\(\frac{0 – –3}{–2– 4} = m\)

\(\frac{3}{–6} = –\frac{1}{2} = m\)

Using the Slope Formula

Here is the formula again:

\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)

y1 is the first y coordinate used in the calculation. y2 is the second y coordinate used in calculation.

In the same way, x1 is the first x coordinate used in the calculation, and x2 is the second x coordinate used in calculation.

m is the variable that is used for the slope of the line.

Free Geometry Review

Slope – Avoiding Common Mistakes

Avoid confusing coordinate 1 with coordinate 2 when substituting values into the slope formula.

Also be sure that your y coordinates are on the top of the fraction and the x coordinates are on the bottom.

You will calculate the slope incorrectly if you mix up the coordinates, so it pays to be careful.

Calculating Slope with Rise and Run – Example

Consider a straight line that has the following coordinates: (2,3) and (6,8).

What is the slope of this line?

How to solve:

First, you need to sort out which values go into each position in the formula.

Remember that coordinates are given in sets such as (x,y).

x and y coordinates

So, the x coordinate is given first in each set of parentheses, followed by the y coordinate.

Here, we have the coordinates (2,3) and (6,8), so y2 = 8 and y1 = 3.

For the x coordinates, x2 = 6 and x1 = 2.

Put the values in the formula

Then, you need to plug the values into the formula to solve.

\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)

\(\frac{rise}{run}= \frac{8 – 3}{6 – 2} = m\)

\(\frac{5}{4} = m\)

Slope – Other Facts

Parallel lines

Parallel lines have the same slope.

Horozontal lines

A horizontal line has a slope of 0.

Vertical lines

A vertical line has no slope. So, we say that the slope of a vertical line is undefined.

Positive slope

A line that points upwards to the right has a positive slope.

Negative Slope

A line that points backwards to the left has a negative slope.

Slope – Further Problems

We will look at x and y intercepts and the point-slope formula in other posts.

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What is the Point Slope Formula?

The point slope formula is the equation of a straight line in the form:

y – y₁ = m(x – x₁)

In the point slope formula, y is the unknown y coordinate. y₁ is the given y coordinate, which is to be placed in the formula.

Similarly, x is the unknown x coordinate. x₁ is the given x coordinate, which is to be used in the formula.

Variable m represents the slope of the straight line.

Setting Up the Point Slope Formula

You can set up the point slope formula if you have the coordinates of one point on the line and the slope of the line.

We have already looked at slope in our post entitled What is Slope?

If you know this formula, go to the Free Geometry Review Problems

Now consider the following example:

You have a straight line with a slope of ¹/₂ and which passes through the point (6,2).  Set up the point slope formula and the equation of the straight line.

First, we need to set up the point slope formula.

To set up the point slope formula, we need to put the information into the formula provided above.

y – y₁ = m(x – x₁)

y – 2 = ¹/₂(x – 6)

Then start to find the equation of the straight line by multiplying each side of the equation by 2 to get rid of the fraction.

y – 2 = ¹/₂(x – 6)

2 × (y – 2) = 2 × [¹/₂(x – 6)]

2y – 4 = x – 6

Then get the x variable to the other side of the equation.

2y – 4 = x – 6

2y – 4 – x = x – x – 6

2y – 4 – x = – 6

The next step is to deal with the integers.

2y – 4 – x = – 6

2y – 4 – x + 6 = – 6 + 6

2y – 4 – x + 6 = 0

Then rearrange the terms to get the formula for a straight line. The formula for a straight line is in this form: ax ± by ± c = 0

2y – 4 – x + 6 = 0

2y – x + 6 – 4 = 0

2y – x + 2 = 0

–x + 2y + 2 = 0

If you need to refresh your memory on the formula for a straight line, please see our post on solving linear equations.

Exam Questions on Point Slope Formula

You might see two different types of problems on the point slope formula on the exam.

Instructions for test problems might be worded like the following examples:

Set up the equation in point slope form.

Find the equation of a straight line.

You can also visit Purple Math for more information on the Point Slope form.

What is the Slope Intercept Form?

The slope intercept form is a formula that identifies the slope and the y intercept of a straight line.

From our post entitled “What is Slope?”, you will remember that the slope measures how steep the line is.

The y intercept is the place where the line crosses the y axis.

You can also think of the y intercept as the value of y when x equals zero.

Slope Intercept Formula

The formula is as follows:

Slope intercept form

y = mx + b

In the slope-intercept formula, these variables are used:

y = the y coordinate

m = slope

x = the x coordinate

b = the y intercept

Example of the Slope Intercept Form

Now have a look at the following example.

y = 3x + 4

The slope of the line is 3 and the y intercept is 4.

The y intercept tells us that the point (0,4) lies on this line.

Putting Other Equations into the Slope Intercept Form

An equation of a single line can take various forms.

As you know, you can solve linear equations by substituting values in the equation.

In addition, you can get to the slope-intercept form by changing the standard equation of a line.

For instance, consider the equation of a line:

–2x + 5y = 10

We can get the above equation into the slope-intercept form in three steps.

First, isolate the y term to the left:

–2x + 5y = 10

5y = 10 + 2x

Then divide by 5 to solve for y:

5y = 10 + 2x

5y ÷ 5 = (10 + 2x) ÷ 5

y = 2 + ^2x/₅

Finally, reorganize the right side of the equation to get the slope-intercept form.

y = 2 + ^2x/₅

y = ^2x/₅ + 2

Graphing with Slope-Intercept

Let’s look at the previous equation again.

y = ^2x/₅ + 2

We can use this equation in order to graph the line.

First, plot the y intercept, which is point (0,2).

Then use the slope to count the spaces to the next point.

The slope in our example is ²/₅.

Remember that the change in the y coordinate is given in the top of the fraction for the slope formula.

The change in the x coordinate is given in the bottom of the fraction

In other words, the top number in the fraction shows how many units we need to move to up.

The bottom number in the fraction shows how many units we need to go to the right.

So, the next point is five units to the right and two units higher than the present point.

FREE GEOMETRY REVIEW QUESTIONS

Questions on Solving Linear Equations

Questions on solving linear equations may be some of the easiest equation questions that you will see on your exam.

Some of the simplest questions on linear equations will involve only addition or subtraction.

A very basic question may look like the following example:

x + y + 3 = 6

To solve the above equation, you need to isolate x to one side of the equation.

Now look at the solution that follows.

x + y + 3 = 6

First, deal with the integer.

x + y + 3 = 6

x + y + 3 − 3 = 6 − 3

Then perform the operations.

x + y + 3 − 3 = 6 − 3

x + y = 3

Finally, isolate x to find the solution for this basic linear equation.

x + y = 3

x + y − y = 3 − y

x = 3 − y

General Equation of a Straight Line

The following equation can be used for any straight line:

ax + by + c = 0

We can deduce a couple of important points from this equation.

First of all, we can put in 0 for y in order to find the point where the line meets the x axis.

This is called the x intercept.

Equation for the x intercept

ax + by + c = 0

ax + 0y + c = 0

ax + c = 0

ax + c − c = 0 − c

ax = −c

\(x = \frac{-c}{a}\)

Secondly, we can put in 0 for x in order to find the point where the line meets the y axis.

This is called the y intercept.

Equation for the y intercept

ax + by + c = 0

a0 + by + c = 0

by + c = 0

by + c − c = 0 − c

by = −c

\(y = \frac{-c}{b}\)

Plotting Linear Equations

Now look at how we can plot a line on the coordinate plane.

Consider this linear equation:  3x − 4y + 3 = 0

First, find the point at which the line intersects the x axis by substituting 0 for y.

3x − 4y + 3 = 0

3x − 0y + 3 = 0

3x + 3 = 0

3x + 3 − 3 = 0 − 3

3x = −3

x = −1

So when y = 0, x = −1

Next, find the point at which the line intersects the y axis by substituting 0 for x.

3x − 4y + 3 = 0

0x − 4y + 3 = 0

−4y + 3 = 0

−4y + 3 − 3 = 0 − 3

−4y = −3

\(y = \frac{3}{4}\)

We know both the x and y intercepts at this point.

So, we can plot the linear equation as shown below:

We look at linear equations in more depth in our posts on graphing.

Free Geometry Review Questions