Find the equation of a line that passes through the two points (-3,3) and (0,6)

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First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow.

The Points

We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is:

Steps

There are 3 steps to find the Equation of the Straight Line :

• 1. Find the slope of the line
• 2. Put the slope and one point into the "Point-Slope Formula"
• 3. Simplify

Step 1: Find the Slope (or Gradient) from 2 Points

What is the slope (or gradient) of this line?

We know two points:

• point "A" is (6,4) (at x is 6, y is 4)
• point "B" is (2,3) (at x is 2, y is 3)

The slope is the change in height divided by the change in horizontal distance.

Looking at this diagram ...

Slope m  =  change in ychange in x  =  yA − yBxA − xB

In other words, we:

• subtract the Y values,
• subtract the X values
• then divide

Like this:

m  =   change in y change in x   =   4−3 6−2   =   1 4 = 0.25

It doesn't matter which point comes first, it still works out the same. Try swapping the points:

m  =   change in y change in x   =   3−4 2−6   =   −1 −4 = 0.25

Same answer.

Step 2: The "Point-Slope Formula"

Now put that slope and one point into the "Point-Slope Formula"

Start with the "point-slope" formula (x1 and y1 are the coordinates of a point on the line):

y − y1 = m(x − x1)

We can choose any point on the line for x1 and y1, so let's just use point (2,3):

y − 3 = m(x − 2)

We already calculated the slope "m":

m = change in ychange in x = 4−36−2 = 14

And we have:

That is an answer, but we can simplify it further.

Step 3: Simplify

Start with:y − 3 = 14(x − 2)

Multiply 14 and (x−2):y − 3 = x4 − 24

Add 3 to both sides:y = x4 − 24 + 3

Simplify:y = x4 + 52

And we get:

Which is now in the Slope-Intercept (y = mx + b) form.

Let us confirm by testing with the second point (6,4):

y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4

Yes, when x=6 then y=4, so it works!

Another Example

The Big Exception

The previous method works nicely except for one particular case: a vertical line:

Copyright © 2017 MathsIsFun.com

The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula. To find the equation of a line for any given two points that this line passes through, use our slope intercept form calculator.

1. Identify the coordinates (x₁,y₁)and(x₂,y₂). We will use the formula to calculate the slope of the line passing through the points (3,8) and (-2, 10).
2. Input the values into the formula. This gives us (10 - 8)/(-2 - 3).
3. Subtract the values in parentheses to get 2/(-5).
4. Simplify the fraction to get the slope of -2/5.
5. Check your result using the slope calculator.

To find the slope of a line we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the rise, divided by the change of the x-coordinate, known the the run. The calculations in finding the slope are simple and involves nothing more than basic subtraction and division.

slope = (y₂ - y₁) / (x₂ - x₁)

Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values.

It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula.

Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated. The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with the its vertices touching the midpoint of the sides of the larger polygon. This can be obtained using the midpoint calculator or by simply taking the average of each x-coordinates and the average of the y-coordinates to form a new coordinate.

The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator. You can also use the distance calculator to compute which side of a triangle is the longest, which helps determine which sides must form a right angle if the triangle is right.

The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to lower right, then the gradient is negative.

The method for finding the slope from an equation will vary depending on the form of the equation in front of you. If the form of the equation is y=mx+c, then the slope (or gradient) is just m. If the equation is not in this form, try to rearrange the equation. To find the gradient of other polynomials, you will need to differentiate the function with respect to x.

1. Use a map to determine the distance between the top and bottom of the hill as the crow flies.
2. Using the same map, or GPS, find the altitude between the top and bottom of the hill. Make sure that the points you measure from are the same as step 1.
3. Convert both measurements into the same units. Use the Omni Length Converter if you are unsure.
4. Divide the difference in altitude by the distance between the two points.
5. This number is the gradient of the hill if it increases linearly. If it does not, repeat the steps but at where there is a noticeable change in slope.

1. Measure the difference between the top and bottom of the slope in relation to both the x and y axis.
2. If you can only measure the change in x, multiply this value by the gradient to find the change in the y axis.
3. Make sure the units for both values are the same.
4. Use Pythagoras’ theorem to find the length of the slope. Square both the change in x and the change in y.
5. Add the two values together.
6. Find the square root of the summation.
7. This new value is the length of the slope.

A 1/20 slope is one that rises by 1 unit for every 20 units traversed horizontally. So, for example, a ramp that was 200 ft long and 10 ft tall would have a 1/20 slope. A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis.

As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you’d like to find the gradient.

The rate of change of a graph is also its slope, which are also the same as gradient. Rate of change can be found by dividing the change in the y (vertical) direction by the change in the x (horizontal) direction, if both numbers are in the same units, of course. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa).

Slopes (or gradients) have a number of uses in everyday life. There are some obvious physical examples - every hill has a slope, and the steeper the hill, the greater its gradient. This can be useful if you are looking at a map and want to find the best hill to cycle down. You also probably sleep under a slope, a roof that is. The slope of a roof will change depending on the style and where you live. But, more importantly, if you ever want to know how something changes with time, you will end up plotting a graph with a slope.

A 10% slope is one that rises by 1 unit for every 10 units travelled horizontally (10 %). For example, a roof with a 10% slope that is 20 m across will be 2 m high. This is the same as a gradient of 1/10, and an angle of 5.71° is formed between the line and the x-axis.

To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. For linear equations:

1. Put the equation into the form y=mx+c.
2. Write a new line where you add 1 to the order of the x (e.g., x becomes x^2, x^2.5 becomes x^3.5).
3. Divide m by the new number of the order and put it in front of the new x.
4. Multiply c by x and add this to the new line.
5. Solve this new line twice, one where x is the upper bound of the area you wish to find and one where x is the lower bound.
6. Subtract the lower bound from the upper bound.
7. Congratulate yourself on your achievement.

A 5 to 1 slope is one that, for every increase of 5 units horizontally, rises by 1 unit. The number of degrees between a 5 to 1 slope and the x-axis is 11.3°. This can be found by first calculating the slope, by dividing the change in the y direction by the change in the x direction, and then finding the inverse tangent of the slope.