Equation of a Line

y = mx + b

In the above coordinate planes, the lines appeared without any explanation as to why the line pointed in a certain direction at a certain steepness. The location and slant of a line are determined by an equation. It is the line that graphs out all the points that satisfy this equation. Since this is important, it bears repeating: a line on a coordinate plane is a graphical representation of a series of points that fulfill a mathematical equation.

The standard form in which linear equations which are graphed appear on a coordinate plane is:

y = mx + b

• y is the y-coordinate (or the number of spaces vertically above or below the x-axis)
• m is the slope, or the degree of steepness of the line, as defined above
• x is the x-coordinate (or the number of spaces horizontally right or left from the y-axis)
• b is the y-intercept, which is the number of units above or below the horizontal axis where the line crosses the vertical axis

Consider the following example:

Graph the series of points satisfying the equation y = 2x + 3

One means to do this would be to manually generate a list of points that satisfy this equation. For example, if x = 0, y must equal 3; if x = 1, y must equal 5; etc.

However, there is a faster way. According to the y = mx + b formation of a line, m = 2 and b = 3. Consequently, the line being graphed must cross the vertical axis 3 units above the horizontal axis and it must rise vertically 2 units for every 1 unit it runs horizontally.

Writing the Equation of a Line

It is important to know how to take a pair of points (whether from a graph or from a word problem) and write an equation for a line that satisfies the two points. This process involves solving for m and b in the equation y = mx + b. Consider the following example:

Find the equation of a line that passes through the points (3, 6) and (-2, -4).

The goal is to take these two points and write an equation in the form y = mx + b that passes through the points.

• Find m, the slope
  Slope = rise/run = (6 - [-4])/(3 - [-2]) = 10/5 = 2
• Plug in a point (it does not matter which one) and solve for b
  y = 2x + b
  6 = 2(3) + b
  6 = 6 + b
  b = 0
• Plug in m and b to write an equation:
  y = 2x

Axis Intercepts

The x and y-intercept are important properties of a line and it is often necessary to find the exact location where a line intersects the x-axis and y-axis. The best means to find an intercept is algebraically.

X-Axis Intercept

When a line crosses the x-axis, its y-value will be zero. Consequently, by setting y = 0 and solving for x, the x-coordinate at which the line crosses the x-axis can be found.

What is the x-intercept of the equation y = 2x + 4?

The line will cross the x-axis at y = 0, or ordered pair (x, 0).

y = 0 = 2x + 4

-4 = 2x

x = -2

The line will cross the x-axis at x = -2 and y = 0.

Y-Axis Intercept

When a line crosses the y-axis, its x-value will be zero. As a result, by setting x = 0 and solving for y, the y-coordinate at which the line crosses the y-axis can be found. If an equation is in y = mx+b format, recall that since setting x = 0 yields y = b, the value of b is the y intercept.

What is the y-intercept of the equation y = -15x + 17?

The line will cross the y-axis at x = 0, or ordered pair (0, y).

y = -15(0) + 17

y = 0 + 17

y = 17

The line will cross the y-axis at x = 0 and y = 17.

Parallel Lines

Parallel lines are lines that never intersect. In order to never intersect, two lines must have the same angle (technically called slope). If two lines do not have the same slope, they will eventually intersect. However, if two distinct lines have the same slope, they will never intersect.

What is the slope of a line parallel to the line that connects points (1, 4) and (5, 16)?

The line that is parallel will have the same slope as the line that connects the two points mentioned above.

Slope of line connecting two points: rise/run = (16-4)/(5-1) = 12/4 = 3

Consequently, any line with a slope of 3 will be parallel with the line that connects (1, 4) and (5, 16).

Perpendicular Lines

Line A is perpendicular to Line B if Line A intersects Line B at a 90° angle. The most important property of perpendicular lines is as follows:

The slope of perpendicular lines forms negative reciprocals

Distance Between Points

In order to find the distance between two points, either: (1) use the distance formula [to be derived] or (2) draw a triangle and use the Pythagorean theorem.

The distance formula comes from the Pythagorean theorem, as the example below shows:

In the graph below, what is the distance between points K and L?

The best means to solve this type of a question is by drawing in a triangle and solving for the hypotenuse, which is the distance between points K and L. In order to do this, sketch in a triangle by placing a third point such that a right angle is formed (point N below is such a point):

By inspection, the location of each point is as follows:

L: (2, 1)

N: (5, 1)

K: (5, 5)

Find the length of each leg of the right triangle:

LN = 5 - 2 = 3

KN = 5 - 1 = 4

Use the Pythagorean theorem to find the length of KL:

(KL)² = (LN)² + (KN)² =

(KL)² = 3² + 4²

(KL)² = 9 + 16

(KL)² = 25

KL = 5

Distance Formula

The process above can be simplified using the following formula:

The coordinates of K are x1, y1 or (5, 5).

The coordinates of L are x2, y2 or (2, 1).

d = KL

Notice that the distance formula immediately above is simply a formulaic representation of the graphical process undertaken above to solve for KL.