Question

t - shirt business, system of equations
name

1. after you sell out of your first batch of t - shirts, you decide to make a second batch. this time around you narrow it down to these two options:

option x: 100 cotton t - shirts for $360. now that you have some money and a customer base built up, you can afford to sell these for only $10 each.
option y: 100 poly - blend shirts for $460. these shirts are nicer than the cotton shirts, but your research tells you that, due to some troubles in the local economy, youll only be able to sell them for $10 each.
create an equation in slope - intercept form (y = mx + b) for both options.
option x equation
y = 10x - 360
option y equation
y = 10x - 460

1. set the right sides of these equations equal to each other, and solve for x.
2. what type of solution do you get? what does this mean about which option is more profitable?

Explanation:

Step1: Set equations equal

Given Option X: \( y = 10x - 360 \) and Option Y: \( y = 10x - 460 \). Set \( 10x - 360 = 10x - 460 \).

Step2: Solve for x

Subtract \( 10x \) from both sides: \( -360 = -460 \). This is a contradiction, so no solution.

Answer:

No solution (parallel lines, same slope different y - intercepts). This means the profit functions never intersect; since Option X has a higher y - intercept (\(-360\) vs \(-460\)) with same slope, Option X is always more profitable.