Question

for each equation, determine whether it is linear.
equation | is the equation linear?
| yes | no
(a) $y = x$ | $\circ$ | $\circ$
(b) $y = -4x + 7$ | $\circ$ | $\circ$
(c) $y = -6$ | $\circ$ | $\circ$
(d) $y = 2x^2 + 5$ | $\circ$ | $\circ$

Explanation:

Part (a)

Step1: Recall linear equation form

A linear equation in two variables is of the form \( y = mx + b \), where \( m \) (slope) and \( b \) (y - intercept) are constants, and the highest power of the variable \( x \) is 1.
For the equation \( y=x \), we can rewrite it as \( y = 1\times x+0 \). Here, \( m = 1 \) and \( b = 0 \), and the power of \( x \) is 1.

Step2: Determine linearity

Since it fits the form of a linear equation \( y=mx + b \) with the highest power of \( x \) being 1, the equation \( y = x \) is linear.

Part (b)

Step1: Recall linear equation form

A linear equation in two variables is of the form \( y=mx + b \), where \( m \) (slope) and \( b \) (y - intercept) are constants, and the highest power of the variable \( x \) is 1.
The equation \( y=- 4x + 7 \) is in the form \( y=mx + b \) where \( m=-4 \) and \( b = 7 \), and the power of \( x \) is 1.

Step2: Determine linearity

Since it fits the form of a linear equation \( y = mx + b \) with the highest power of \( x \) being 1, the equation \( y=-4x + 7 \) is linear.

Part (c)

Step1: Recall linear equation form

A linear equation in two variables can also be written as \( y=mx + b \). If we rewrite \( y=-6 \) as \( y=0\times x-6 \), here \( m = 0 \) (which means it is a horizontal line) and \( b=-6 \), and the power of \( x \) is 1 (since the term with \( x \) has a coefficient of 0).

Step2: Determine linearity

Since it can be written in the form \( y=mx + b \) with the highest power of \( x \) being 1, the equation \( y = - 6 \) is linear.

Part (d)

Answer:

s:
(a) For \( y = x \), select "Yes" (the circle under "Yes" for this row).
(b) For \( y=-4x + 7 \), select "Yes" (the circle under "Yes" for this row).
(c) For \( y=-6 \), select "Yes" (the circle under "Yes" for this row).
(d) For \( y = 2x^{2}+5 \), select "No" (the circle under "No" for this row).