Math

A linear function with an equation written in the form $y = m x + b$ can be expressed in the form $A x + B y = C$ and vice versa.

Example 1

Express each equation in the form $A x + B y = C$

y = - 2 x + 5

y = - 5

y = \frac{1}{3} x - 1

Solution

a. Use the addition property of equality.

The required form is $2 x + y = 5,$ where $A = 2,$ $B = 1,$ and $C = 5.$

b. The given equation $y = - 5$ is already in the form $A x + B y = C,$ where $A = 0,$ $B = 1,$ and $C = - 5.$

c. Apply the properties of equality.

y	=	$\frac{1}{3} x - 1$	Given
$y - \frac{1}{3} x$	=	$\frac{1}{3} x - 1 - \frac{1}{3} x$	Subtract $\frac{1}{3} x$ from both sides.
$y - \frac{1}{3} x$	=	−1	Simplify.
−3y + x	=	3	Multiply both sides by –3.
x + 3y	=	3	Rearrange the terms.

Thus, the required form is $x - 3 y = 3,$ where $A = 1,$ $B = - 3,$ and $C = 3.$