Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(a) $7xy + 4x = 8$	$\circ$	$\circ$
(b) $x = 8y^3 + y$	$\circ$	$\circ$
(c) $6x = 3 - y$	$\circ$	$\circ$
(d) $\frac{4}{x} - 8y = 5$	$\circ$	$\circ$

Explanation:

Step1: Recall linear equation rule

A linear equation in two variables has the form $Ax + By = C$, where $A,B,C$ are constants, and variables have degree 1, no variable products/reciprocals.

Step2: Analyze equation (a)

Equation: $7xy + 4x = 8$
It has a term $7xy$ (product of variables, degree 2). Not linear.

Step3: Analyze equation (b)

Equation: $x = 8y^3 + y$
It has $y^3$ (variable to degree 3). Not linear.

Step4: Analyze equation (c)

Equation: $6x = 3 - y$
Rearrange to $6x + y = 3$, fits $Ax+By=C$. Linear.

Step5: Analyze equation (d)

Equation: $\frac{4}{x} - 8y = 5$
$\frac{4}{x}=4x^{-1}$ (variable to negative degree). Not linear.

Answer:

(a) No
(b) No
(c) Yes
(d) No