Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(a) $y = x - 8$	$circ$	$circ$
(b) $y = 6x$	$circ$	$circ$
(c) $y = -x^3$	$circ$	$circ$
(d) $y = 3x^2 - 4$	$circ$	$circ$

Explanation:

Step1: Recall linear equation form

A linear equation in two variables follows the form $y=mx+b$, where $m$ and $b$ are constants, and the exponent of $x$ is 1.

Step2: Analyze equation (a)

Equation: $y=x-8$
This matches $y=mx+b$ with $m=1$, $b=-8$; $x$ has exponent 1.

Step3: Analyze equation (b)

Equation: $y=6x$
This matches $y=mx+b$ with $m=6$, $b=0$; $x$ has exponent 1.

Step4: Analyze equation (c)

Equation: $y=-x^3$
Here, $x$ has exponent 3, which violates the linear form.

Step5: Analyze equation (d)

Equation: $y=3x^2-4$
Here, $x$ has exponent 2, which violates the linear form.

Answer:

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