Composing a linear formula

Linear formulas and equations: Linear functions

Composing a linear formula

	Procedure
	With a graph or table of a linear formula, we can compose a formula of the form #y=\blue a \cdot x +\green b# in the following manner.
Step 1	Determine intercept #\green b# by seeing which #y#-value corresponds with #x=0#.
Step 2	Pick two "nice" or convenient points #A# and #B# with coordinates #\rv{x_A, y_A}# and #\rv{x_B, y_B}#.
Step 3	Calculate slope #a# with \[\blue a=\frac{y_B-y_A}{x_B-x_A}\]
Step 4	Enter the found #\blue a# and #\green b# in the formula #y=\blue a \cdot x +\green b#.

Intercept is not readable

It is not always possible to see which #y#-value corresponds with #x=0#. In those cases, another way to calculate the value #\green{b}# is by using two points on the line. We will explain that in Composing the equation of a line.

Create the formula of the linear formula corresponding to the following graph.

The formula is equal to #y=-3 \cdot x + 4#.

We can calculate this as follows.
Step 1: The intercept #b# is the #y#-value of the intersection point of the line and the #y#-axis. In this case, it is #4#.

Step 2: We choose two grid points, for example, #A# with coordinates #\rv{0,4}# and #B# with coordinates #\rv{2,-2}#

Step 3: We now calculate slope #a#. Here, #a=\tfrac{y_B-y_A}{x_B-x_A}=\tfrac{-2-4}{2-0}=\tfrac{-6}{2}=-3#

Step 4: We can now substitute the found values for #a# and #b# in the formula #y=a \cdot x+b#. The formula, therefore, becomes #y=-3 \cdot x + 4#.

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