Question

for each equation, determine whether it is linear.

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

Explanation:

Step1: Recall linear equation definition

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

Step2: Analyze equation (a) \( y = x^3 \)

The power of \( x \) is 3, which is not 1. So it's not linear. Mark "No".

Step3: Analyze equation (b) \( y = x^2 - 5 \)

The power of \( x \) is 2, not 1. Not linear. Mark "No".

Step4: Analyze equation (c) \( y = -x + 2 \)

This is in the form \( y = mx + b \) ( \( m = -1 \), \( b = 2 \) ), highest power of \( x \) is 1. Linear. Mark "Yes".

Step5: Analyze equation (d) \( y = -7x \)

This is in the form \( y = mx + b \) ( \( m = -7 \), \( b = 0 \) ), highest power of \( x \) is 1. Linear. Mark "Yes".

Answer:

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

(For the table, mark the circles as follows:
(a) No column's circle.
(b) No column's circle.
(c) Yes column's circle.
(d) Yes column's circle.)