The cost y for a company to produce x​ T-shirts is given by the equation y=15x+1500​, and the revenue y from the sale of these​ T-shirts is y=30x. Find the​ break-even point, the point where the line that represents the cost intersects the revenue line?

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Oct 6, 2017

$\left(100 , 3000\right)$

Explanation:

Essentially, this problem is asking you to find the intersection point of these two equations. You can do this by setting them equal to each other, and since both equations are written in terms of y, you don't have to do any preliminary algebraic manipulation:

$15 x + 1500 = 30 x$

Let's keep the $x ' s$ on the left-hand side and the numerical values on the right-hand side. To achieve this goal, subtract $1500$ and $30 x$ from both sides:

$15 x - 30 x = - 1500$

Simplify:

$- 15 x = - 1500$

Divide both sides by $- 15$:

$x = 100$

Careful! This isn't the final answer. We need to find the POINT where these lines intersect. A point is comprised of two components - it's x coordinate and it's y coordinate. We found the x coordinate, so now all we have to do is plug in $x = 100$ into any of the two original equations to find the y coordinate. Let's use the second one:

$y = 30 x$
$y = 30 \cdot 100 = 3000$

So the point of intersection is $\left(100 , 3000\right)$.