Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(a) $y = 6x$	$\circ$	$\circ$
(b) $y = x - 6$	$\circ$	$\circ$
(c) $y = 2x^2 - 1$	$\circ$	$\circ$
(d) $y = -x^3$	$\circ$	$\circ$

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 highest power of \(x\) is 1.

Step2: Analyze equation (a) \(y = 6x\)

This can be written as \(y = 6x + 0\), where \(m = 6\) and \(b = 0\). The power of \(x\) is 1. So, it is linear. Choose "Yes" for (a).

Step3: Analyze equation (b) \(y = x - 6\)

This is in the form \(y = mx + b\) with \(m = 1\) and \(b = -6\). The power of \(x\) is 1. So, it is linear. Choose "Yes" for (b).

Step4: Analyze equation (c) \(y = 2x^2 - 1\)

The highest power of \(x\) is 2 (from \(x^2\)), which is not 1. So, it is not linear. Choose "No" for (c).

Step5: Analyze equation (d) \(y = -x^3\)

The highest power of \(x\) is 3 (from \(x^3\)), which is not 1. So, it is not linear. Choose "No" for (d).

Answer:

• (a) \(y = 6x\): Yes
• (b) \(y = x - 6\): Yes
• (c) \(y = 2x^2 - 1\): No
• (d) \(y = -x^3\): No