Question

which equations describe a linear function? select all that apply. select all correct options 3y = x; y = x³; y = 5x² + 6x; x - 3y = 12; 4x + 3y² = 24

Explanation:

Step1: Recall linear function form

A linear function has the form \( y = mx + b \) (or can be rewritten into this form), where the highest power of \( x \) and \( y \) is 1.

Step2: Analyze \( 3y = x \)

Rewrite \( 3y = x \) as \( y=\frac{1}{3}x + 0 \). Here, the power of \( x \) is 1 and power of \( y \) is 1. So it's linear.

Step3: Analyze \( y = x^{3} \)

The power of \( x \) is 3, which is greater than 1. So it's not linear (it's a cubic function).

Step4: Analyze \( y = 5x^{2}+6x \)

The highest power of \( x \) is 2, so it's a quadratic function, not linear.

Step5: Analyze \( x - 3y = 12 \)

Rewrite as \( - 3y=-x + 12 \), then \( y=\frac{1}{3}x-4 \). Power of \( x \) and \( y \) is 1, so linear.

Step6: Analyze \( 4x + 3y^{2}=24 \)

The power of \( y \) is 2, so it's a quadratic (in terms of \( y \)) function, not linear.

Answer:

\( 3y = x \), \( x - 3y = 12 \) (corresponding to the green and blue - colored option boxes respectively)