How to Find B in Y=Mx + B

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The formula ​y​ = ​mx​ + ​b​ is an algebra classic. It represents a linear equation, the graph of which, as the name suggests, is a straight line on the ​x​-, ​y​-coordinate system.

Often, however, an equation that can ultimately be represented in this form appears in disguise. As it happens, any equation that can appear as:

Ax + By = C

where ​A​, ​B​ and ​C​ are constants, ​x​ is the independent variable and ​y​ is the dependent variable is a linear equation. Note that ​B​ here is not the same as ​b​ above.

The reason for recasting it in the form

y = mx + b

is for ease of graphing. ​m​ is the slope, or tilt, of the line on the graph, whereas ​b​ is the ​y​-intercept, or the point (0. ​y​) at which the the line crosses the ​y​, or vertical, axis.

If you already have an equation in this form, finding ​b​ is trivial. For example, in:

y = -5x -7

All terms are in the proper place and form, because ​y​ has a ​coefficient​ of 1. The slope ​b​ in this instance is simply −7. But sometimes, a few steps are required to get there. Say you have an equation:

6x - 3y = 21

To find ​b​:

Step 1: Divide All Terms in the Equation by B

This reduces the coefficient of ​y​ to 1, as desired.

\frac{6x - 3y}{3} = \frac{21}{3} \\ \,\\ 2x - y = 7

Step 2: Rearrange the Terms 

For this problem:

-y = 7 + 2x \\ y = -7 - 2x \\ y = -2x -7 \\

The ​y​-intercept, ​b​ is therefore ​−7​.

Step 3: Check the Solution in the Original Equation

Inserting the result with ​x​ = 0:

6x -3y = 21 \\ (6 × 0) - (3 × -7) = 21 \\ 0 + 21 = 21

The solution, b = −7, is correct.

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