Applying Systems of Linear Equations to Market Equilibrium: Steps & Example Businesses use market equilibrium to determine price and sell products. Learn how to use systems of linear equations to nd market equilibrium in this video lesson. Review what you know with a short quiz!
LINEAR MOELS AN LEMONS Max is a lemonade millionaire with his own lemonade stand company, Xtreme Lemo Lemon. n. Max Max need needs s to unde unders rsta tand nd supp su pply ly and and dema demand nd so he can can fnd fnd market equilibrium. Market equilibrium is when when the the amou amount nt o pro product duct pro produc duced is equa equall to the the amount o quantity demanded. We can see equilibrium on a graph when the supply unction and the demand unction intersect, like shown on this graph. Max can then fgure out how to price his new lemonade products based on market equilibrium. equilibrium. Lets break this down one line at a time.
MAR!E" E#$ILI%RI$M !his is the line o the supply unction. "n the graph, the y #axis #axis represents the price o a product, while the x #axis #axis represents the quantity o the product. $o i we put a point at %&', &'( on this line, that would tell us that Max is willing to supply &' cups o lemonade i he can charge )&' or each cup. !his is a pretty ancy cup o lemonade i you ask me. Lets say that Max has changed his mind and is now willing to supply &*.+ cups o lemonade, but only i the price o the lemonade is )+ per cup. e is also willing to supply -' cups o lemonade or )./+ per cup. We can put that inormation into two points on a graph like this0 '()*+ *, an- ./+ 0)1*,) 1ow we ha2e two points but no line. We need to write an equation to fnish this supply unction. !o do this, you need to know the slope ormula, which y ( 3 y ', 4 x ( 3 x 3 x ',, and the linear equation is y 2 2 mx 5 5 b) is m 2 y
6ININ7 "8E S$99L 6$N;"ION &. 3ind the slope0 m 2 y ( 3 y ', 4 x ( 3 x ',. a. 4lug in the numbers &*.+, +, -', and ./+ and e2aluate the equation0 m 5 %./+ # +( 6 %-' # &*.+( m 5 &./+ 6 &/.+ m 5 '.& "kay, so or this equation we know that slope is '.&, <=i>= is t=e same as a '/? in>rease in pri>e . 1ow, we need to fnd the y #intercept o the equation. We will use the point# slope orm to fnd the y #intercept. We can use our smallest numbered point %&*.+, +( and our slope, '.&, to sol2e the equation, although you can use which e2er point you choose. 4lug the numbers into the equation and e2aluate0 % y # +( 5 '.&% x # &*.+( 7se the distributi2e property. y # + 5 '.& x # &.*+ 8dd like terms. y 5 '.& x 9 -./+ 1ow we know that the supply unction or Maxs Xtreme Lemon product is y 2 /)' x 5 .)1*. :asically, the .)1* represents t=e o@erall >ost of making t=e pro-u>t or t=e lo
e t=at t=e pro-u>t >an be supplie- . 8lso, you would normally replace the y with a p to indicate the price o the product and x with a q to indicate the quantity o the product supplied. ; let the y and x in while we were working to pre2ent conusion, but you really need to get used to seeing supply unctions written like this0 p 2 /)'q 5 .)1*. ;t is also good to note here that t=e slope of t=e supply fun>tion
6ININ7 "8E EMAN 6$N;"ION Max conducts a sur2ey to fnd the demand unction or his consumer audience. e fnds that the consumers are willing to buy ' cups o lemonade i the lemonade is )< a cup. !hey are also willing to pay )/ per cup o lemonade, but they will only buy -' cups. !his makes sense because most
people are more likely to buy a lot o a product i it is less expensi2e. ;n act, the la< of -eman- states the quantity demanded will increase as the price o the product decreases. !ry pausing the 2ideo and work the equations as we go= Lets use the slope ormula frst to fgure out the rate o this equation. 4lug the numbers into the slope ormula and e2aluate. My points are 0/+ , an./+ 1,: m 5 y ( 3 y ', 4 x ( 3 x ',. m 5 %/ # <( 6 %-' # '( m 5 - 6 #-' m 5 #'.&
