Question

for each equation, determine whether it is linear.
equation | is the equation linear?
| yes | no
(a) $y = x^2 + 7$ | $\circ$ | $\circ$
(b) $y = x + 8$ | $\circ$ | $\circ$
(c) $y = -3x^3$ | $\circ$ | $\circ$
(d) $y = 4^x$ | $\circ$ | $\circ$

Explanation:

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 equation (a) \( y = x^2 + 7 \)

The exponent of \( x \) is 2, which is not 1. So, it is not linear. Select "No".

Step3: Analyze equation (b) \( y = x + 8 \)

This is in the form \( y = mx + b \) with \( m = 1 \) and \( b = 8 \), and the exponent of \( x \) is 1. So, it is linear. Select "Yes".

Step4: Analyze equation (c) \( y = -3x^3 \)

The exponent of \( x \) is 3, which is not 1. So, it is not linear. Select "No".

Step5: Analyze equation (d) \( y = 4^x \)

This is an exponential function (variable in the exponent), not in the form \( y = mx + b \). So, it is not linear. Select "No".

Answer:

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