Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(a) $y = -x^2 + 7$	$\circ$	$\circ$
(b) $y = 3^x$	$\circ$	$\circ$
(c) $y = 5x^3$	$\circ$	$\circ$
(d) $y = 2x - 6$	$\circ$	$\circ$

Explanation:

Part (a)

Step1: Recall linear equation form

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

Step2: Analyze \( y=-x^{2}+7 \)

In the equation \( y = -x^{2}+7 \), the exponent of \( x \) is 2, which is not 1. So it does not follow the linear equation form.

Step1: Recall linear equation form

A linear equation is of the form \( y=mx + b \) (exponent of \( x \) is 1, no exponential functions of \( x \)).

Step2: Analyze \( y = 3^{x} \)

The equation \( y=3^{x} \) is an exponential function (variable in the exponent), not in the form \( y=mx + b \).

Step1: Recall linear equation form

Linear equations have \( x \) with exponent 1.

Step2: Analyze \( y = 5x^{3} \)

In \( y = 5x^{3} \), the exponent of \( x \) is 3, not 1. So it is not linear.

Answer:

For (a): No (select the "No" option for \( y = -x^{2}+7 \))

Part (b)