Question

for each equation, determine whether it is linear.
equation\tis the equation linear?
\tyes\tno
(a) $y = 2^x$\t$circ$\t$circ$
(b) $y = -x^2 + 9$\t$circ$\t$circ$
(c) $y = 5x^3$\t$circ$\t$circ$
(d) $y = 2x - 6$\t$circ$\t$circ$

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 $x$ has an exponent of 1 (no exponents other than 1, no variable as an exponent).

Step2: Analyze equation (a)

Equation: $y=2^x$
Here, $x$ is the exponent, not a linear term. So it is not linear.

Step3: Analyze equation (b)

Equation: $y=-x^2+9$
The variable $x$ has an exponent of 2, which violates the linear form. So it is not linear.

Step4: Analyze equation (c)

Equation: $y=5x^3$
The variable $x$ has an exponent of 3, which violates the linear form. So it is not linear.

Step5: Analyze equation (d)

Equation: $y=2x-6$
This matches the form $y=mx+b$ (where $m=2$, $b=-6$), with $x$ having an exponent of 1. So it is linear.

Answer:

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