Video: Solving Linear Equations to Find the Break-Even Point

A stuffed animal business has a cost function of 𝐢 = 12𝑋 + 30 and a revenue function 𝑅 = 20𝑋. First, find the break-even point. Then, calculate the corresponding revenue.

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Video Transcript

A stuffed animal business has a cost function of 𝐢 equals 12𝑋 plus 30 and a revenue function of 𝑅 equals 20𝑋. First, find the break-even point. Then, calculate the corresponding revenue.

Our cost function 12𝑋 plus 30 and the revenue equals 20𝑋. What is the break-even point then? At the break-even point, the cost of running your business 𝐢 is exactly equal to the money you bring in. At the break-even point, you haven’t made any money, but you have made your whole investment back. So you haven’t lost any money.

Our break-even point will be the 𝑋-value if we set 𝐢 equal to 𝑅. 12𝑋 plus 30 equals 20𝑋. To solve for 𝑋, we’ll isolate 𝑋. Subtract 12 from either side. 12𝑋 minus 12𝑋 cancels out. 20𝑋 minus 12𝑋 equals eight 𝑋. Again to isolate 𝑋, we divide both sides of the equation by eight. Eight divided by eight equals one and one 𝑋 is the same thing as 𝑋. 30 over eight can be simplified to 15 over four.

What we have now is 𝑋 equals 15 over four. But what does this 15 over four tell us? When 15 over four units are created, the cost and the revenue from those 15 over four units should be the same. First, let’s find the revenue of 15 over four units. 20 times 15 over four equals 75. If 15 over four units are sold, the revenue is 75.

What is the cost of 15 over four units? We plug in 15 over four to our 12𝑋 plus 30. 12 times 15 over four equals 45. 45 plus 30 equals 75. What’s happened here is the cost to make 15 over four units is 75 and the revenue is 75. This is the break-even point.

To make money, you would need to sell more than 15 over four units and if you sold less than 15 over four, you would lose money.

The breaking point 15 over four and the revenue at the breaking point 𝑅 equals 75.

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