Applying Systems of Linear Equations to Market Equilibrium: Steps & Example

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  • 0:01 Linear Models and Lemons
  • 0:43 Market Equilibrium
  • 2:49 Finding the Supply Function
  • 5:23 Finding the Demand Function
  • 9:19 Lesson Summary
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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!

Linear Models and Lemons

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.


Market Equilibrium

This is the line of the supply function.

graph of market equilibreum

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).

Graph with points (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 = (ysub2 - ysub1) / (xsub2 - xsub1), the point-slope form, which is y - ysub1 = m(x - xsub1), 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.

Finding the Supply Function

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 = (ysub2 - ysub1) / (xsub2 - xsub1). 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 = 0.1x - 1.25

Add like terms.

y = 0.1x + 3.75

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

Line of the supply function
Graph with plotted 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 = 0.1q + 3.75. It is also good to note here that the slope of the supply function will always be positive.

Finding the Demand Function

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 = (ysub2 - ysub1) / (xsub2 - xsub1)
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 - ysub1 = m(x- xsub1). Plug the numbers into the equation and evaluate:

y - 4 = -0.1(x - 60)

Use the distributive property.

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