Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(b) $y = x$	$\bigcirc$ (yes)	$\bigcirc$ (no)
(c) $y = 2x^2 + 5$	$\bigcirc$ (yes)	$\bigcirc$ (no)
(d) $y = -4x + 7$	$\bigcirc$ (yes)	$\bigcirc$ (no)

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 highest power of $x$ is 1. A constant equation ($y=b$) is also linear (a horizontal line).

Step2: Analyze equation (a)

$y=-9$ fits $y=0x-9$, so it is linear.

Step3: Analyze equation (b)

$y=x$ fits $y=1x+0$, so it is linear.

Step4: Analyze equation (c)

$y=2x^2+5$ has $x$ raised to the power of 2, which violates the linear form, so it is not linear.

Step5: Analyze equation (d)

$y=-4x+7$ fits the linear form $y=mx+b$, so it is linear.

Answer:

Equation	Is the equation linear?
(b) $y=x$	Yes
(c) $y=2x^2+5$	No
(d) $y=-4x+7$	Yes