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Math 105: Precalculus Algebra14 chapters | 113 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Cathryn Jackson*

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Businesses use market equilibrium to determine price and sell products. Learn how to use systems of linear equations to find market equilibrium in this video lesson. Review what you know with a short quiz!

Max is a lemonade millionaire with his own lemonade stand company, Xtreme Lemon. Max needs to understand supply and demand so he can find market equilibrium. **Market equilibrium** is when the amount of product produced is equal to the amount of quantity demanded. We can see equilibrium on a graph when the supply function and the demand function intersect, like shown on this graph. Max can then figure out how to price his new lemonade products based on market equilibrium. Let's break this down one line at a time.

This is the line of the supply function.

Notice the more Max charges for his lemonade, the more he is able to produce. It makes sense; if Max is selling his lemonade for a lot of money, then he can buy more supplies, like lemons, cups, and sugar, and make even more lemonade. This follows the **law of supply**, which states the quantity supplied for the sale will increase as the price of a product increases.

Let's say that Max has changed his mind and is now willing to supply 12.5 cups of lemonade, but only if the price of the lemonade is $5 per cup. He is also willing to supply 30 cups of lemonade for $6.75 per cup. We can put that information into two points on a graph like this: (12.5, 5) and (30, 6.75).

Now we have two points but no line. We need to write an equation to finish this supply function. To do this, you need to know the slope formula, which is *m* = (*y*sub2 - *y*sub1) / (*x*sub2 - *x*sub1), the point-slope form, which is *y* - *y*sub1 = *m*(*x* - *x*sub1), and the slope-intercept form, which is *y* = *mx* + *b*. If you are unfamiliar with any of these equations or feel like you need to review, pause this video and check out the videos on **linear equations and inequalities**.

We will be writing this equation in slope-intercept form. First, to write this equation, we need to find the slope. We can use the information we have to solve the slope formula: *m* = (*y*sub2 - *y*sub1) / (*x*sub2 - *x*sub1). Plug in the numbers 12.5, 5, 30, and 6.75 and evaluate the equation:

*m* = (6.75 - 5) / (30 - 12.5)*m* = 1.75 / 17.5*m* = 0.1

Okay, so for this equation we know that slope is 0.1, which is the same as a 10% increase in price. Now, we need to find the *y*-intercept of the equation. We will use the point-slope form to find the *y*-intercept. We can use our smallest numbered point (12.5, 5) and our slope, 0.1, to solve the equation, although you can use which ever point you choose. Plug the numbers into the equation and evaluate:

(*y* - 5) = 0.1(*x* - 12.5)

Use the distributive property.

*y* - 5 = 0.1*x* - 1.25

Add like terms.

*y* = 0.1*x* + 3.75

Now we know that the supply function for Max's Xtreme Lemon product is *y* = 0.1*x* + 3.75. We can see this represented on the graph as a line.

Basically, the 3.75 represents the overall cost of making the product or the lowest price that the product can be supplied. Also, you would normally replace the *y* with a *p* to indicate the price of the product and *x* with a *q* to indicate the quantity of the product supplied. I left the *y* and *x* in while we were working to prevent confusion, but you really need to get used to seeing supply functions written like this: *p* = 0.1*q* + 3.75. It is also good to note here that the slope of the supply function will always be positive.

Max conducts a survey to find the demand function for his consumer audience. He finds that the consumers are willing to buy 60 cups of lemonade if the lemonade is $4 a cup. They are also willing to pay $7 per cup of lemonade, but they will only buy 30 cups. This makes sense because most people are more likely to buy a lot of a product if it is less expensive. In fact, the **law of demand** states the quantity demanded will increase as the price of the product decreases. Try pausing the video and work the equations as we go!

Let's use the slope formula first to figure out the rate of this equation. Plug the numbers into the slope formula and evaluate. My points are (60, 4) and (30, 7):

*m* = (*y*sub2 - *y*sub1) / (*x*sub2 - *x*sub1)*m* = (7 - 4) / (30 - 60)*m* = 3 / -30*m* = -0.1

This means that the rate is -10%. The more the product costs, the less the consumers will buy. Now that we have the slope, we need to find the *y*-intercept, which will be the price that the consumers will buy 0 cups of lemonade. We will use the point-slope form to find the *y*-intercept: *y* - *y*sub1 = *m*(*x*- *x*sub1). Plug the numbers into the equation and evaluate:

*y* - 4 = -0.1(*x* - 60)

Use the distributive property.

*y* - 4 = -0.1*x* + 6

Add like terms.

*y* = -0.1*x* + 10

Our final equation for the demand function is *y* = -.1*x* + 10. That means that as the price increases, the demand for the product decreases by 10%. The *y*-intercept tells us that the customers will not buy lemonade if it costs $10 or more. The slope of a demand function will always be negative. We can see this function represented on this graph.

-0.1*x* + 10 = 0.1*x* + 3.75

Evaluate the equation by adding like terms.

10 = 0.2*x* + 3.75

6.25 = 0.2*x*

Divide each side by 0.2.

*x* = 31.25

So this tells us that the equilibrium quantity is 31.25 cups of lemonade. Let's plug this into our supply function to figure out the equilibrium price:

*y* = 0.1(31.25) + 3.75*y* = 6.875

So Max and his consumers can both be happy at 31.25 units of lemonade at roughly $6.88 per cup of lemonade. We can see the equilibrium quantity and price represented on this graph.

It was important for Max to find the market equilibrium of his product. **Market equilibrium** is when the amount of product produced is equal to the amount of quantity demanded. Max was able to find the market equilibrium of his product by defining two functions - the supply function and the demand function.

The **supply function** represents the price and quantity at which Max is willing to sell his lemonade. The **demand function** represents the price and quantity at which the consumers are willing to buy Max's lemonade. Now Max knows exactly how much lemonade to make and how much he can charge for his lemonade.

Watch this lesson, then review it thoroughly if your goal is to:

- Comprehend market equilibrium and the laws of supply and demand
- Recall how to find market equilibrium
- Distinguish between the supply function and the demand function
- Calculate the supply, demand, and equilibrium functions
- Demonstrate market equilibrium on a graph

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Math 105: Precalculus Algebra14 chapters | 113 lessons | 12 flashcard sets

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