Equation of the line in its slope form - intersection

slope-intersectionThe equation of the line in its slope-intersection form is written as:

y = mx + b

where m is the slope and b is the intersection of the y axis. Since we only need to know these two values, it is very easy to find the equation of the line from the graph or vice versa, to draw the graph from the equation.

Definition

We know that the general equation of a line is:

Ax + By + C = 0

If we solve for y, we obtain the equation of the line in its slope-intersection form:

y = - (A / B) x - C / A

y = mx + b

Where m = - (A / B) is the slope and b = - C / A is the y-axis intersection, whose coordinate is (0, b).

Example: Let r = 5x + 3y - 5 = 0, find its slope and the y-intercept.

We can solve the problem in two ways:

  1. Solving for “y” and then taking the given equation to the slope-intersection form:

By clearing and we get:

y = - (5/3) x + 5/3

Thus:

m = - (5/3); b = 5/3

  1. Determining A, B and C and then substituting in the equation of the line in its slope-intersection form:

Since r = 5x + 3y - 5 = 0, then:

A = 5; B = 3; C = - 5

As:

y = - (A / B) x - C / A

The slope will be:

m = - A / B = -5/3

And b will be:

b = - C / A = - (-5/3) = 5/3

From graph to equation

If we have the graph of a line we can write its equation just by identifying the slope and the y-intercept

Example: Find the equation of the line graphed below:

Equation of a line - example 1

We can find the slope using the formula:

m = (y2 - Y1) / (x2 - x1)

From the image we observe that the points (-2, -1) and (1, 5) are on the line, therefore:

m = [5 - (- 1)] / [(1 - (-2)] = 6/3 = 2

Now, we observe that the y-axis intersection is b = 3. Consequently, the equation of the line will be:

y = 2x + 3

Equation of a line - example 2

From equation to graph

We can graph a line from its slope-intersection equation.

Example: graph the line y = - (5/3) x + 2.

From the equation we observe that:

m = - 5/3 and b = 2

1. We start by identifying the point (0, b) = (0, 2) on our graph.

Graph 3

2. Since the slope is m = - 5/3, we count 5 units above the previous point and 3 to the left (we can also count 5 down and 3 to the right), then we plot it on our graph.

Graph 4

3. We join the two previous points and thus obtain the graph of y = - (5/3) x + 2

Graph 5