EDGE Math 8

Linear Equations in x and y

Recall that a linear equation in the variables x and y is an equation that can be written in the form $A x + B y + C = 0,$ where A, B, and C are real numbers such that A and B are not both equal to zero. For example, the following are linear equations in two variables:

5 x - 2 y - 9 = 0

F = \frac{9}{5} C + 32

Example

Which of the following equations is linear in two variables?

y = - 2 x + 13

V = s^{3}

a = 5 + 3 (n - 1)

C = π d

A = π r^{2}

y = 3

Solution

1.	The equation $y = - 2 x + 13$ can be written as $2 x + y - 13 = 0.$ Hence, it is linear in the variables x and y.
2.	The equation $V = s^{3}$ is not linear since the degree of $s^{3}$ is 3.
3.	The equation $a = 5 + 3 (n - 1)$ can be written as $a - 3 n - 2 = 0$ so it is linear in the variables a and n.
4.	Recall that $π$ represents a fixed real number. The equation $C = π d$ can be written as $C - π d = 0,$ which is linear in the variables C and d.
5.	The equation $A = π r^{2}$ is not linear since the degree of $r^{2}$ is 2.
6.	The equation $y = 3$ can be considered as a linear equation in two variables since it can be written as $0 x + y - 3 = 0.$

Observe that in a linear equation $A x + B y + C = 0,$ the degree of each variable is equal to 1. Moreover, the coefficients A and B of the variables cannot be simultaneously equal to 0. Thus, if $A = 0,$ then $B \neq 0$ and the equation can be simplified into $y = - \frac{C}{B} .$ On the other hand, if $B = 0,$ then $A \neq 0$ and the equation can be simplified into $x = - \frac{C}{A} .$