Today, I am going to talk about the topic which I have discussed with many of my students, and is a common math application especially when we do shopping, and somebody asks you, for the per unit price of products which you have bought it in bulk on black Fridays discount time.
What is a Systems of Equations with two variable ?
Imagine that you and your friend went for shopping, and you like perfume bottles and you also liked wristbands while going around the shopping mall. Since it is a shopping season and Christmas is around the corner, you and your friend are in hurry and decide to buy both the things in bulk. You purchased 3 perfume bottle and 1 wristband and it costs you $15. Your friend bought 1 perfume bottle and 2 wristband. it costs her $10. You and her, paid the price at the store and returned home. Now, your mom asks you how much is the price of 1 perfume bottle and the unit price of the wristband because she purchased the same item last weekend for $5 each (and you are in real trouble :)) . Here is when we need our friend MATH :) to find us out if we saved money or not.
A system of equations with 2 variables consists of two equations that contain the same two variables. The general form of a system of equations with 2 variables is:
ax + by = c
dx + ey = f
Here, “x” and “y” represent the variables, while “a,” “b,” “c,” “d,” “e,” and “f” are constants. The goal is to find values of “x” and “y” that satisfy both equations simultaneously.
Lets assume that x represents price per unit of perfume bottle and Y represents price per unit of Wristbands. So , we have
3X + Y = $15
X + 2Y = $10
This blog primarily focuses on whether our friend Math can help us here or not :
Conditions for a system of equations with 2 variables :
The conditions for a system of equations with 2 variables depend on the number of solutions it has:
- One Solution : In this case, the system has a unique solution, which means that there is only one answer for per unit price and we can find it. (condition : a1/a2 ≠ b1/b2)
- No Solution : An inconsistent system has no solution, which means that Math cannot help us, because both friends purchased the exact same quantity without knowing per unit price. (condition : a1/a2 = b1/b2 ≠ c1/c2 )
- Infinite Solution: A dependent system has infinitely many solutions, which means that there could be many solutions for per unit price and how much did they pay during this shopping season is hard to determine. (condition : a1/a2 = b1/b2 = c1/c2)
First find the coefficient of x , y and constant terms – which are a1 , a2 for x in both the equations , similarly b1 and b2 for y and c1 and c2 as constant terms. then find what is a1/a2 , b1/b2 and c1/c2.
- If a1/a2 ≠ b1/b2 , it is one solution, then our friend Math can help us to find the price per unit
- If a1/a2 = b1/b2 ≠ c1/c2 , it is no solution, then our friend Math Can’t help us
- If a1/a2 = b1/b2 = c1/c2, it is infinite solution, then Math can help us, but will find many solution.
Here , a1 = 3 , a2 = 1 , b1 = 1 , b2 = 2 , c1 = 15 and c2 =10
a1/a2 = 3 , b1/b2 = 0.5 and c1/c2 = 1.5 . Therefore, we will have one solution.
Once we identify that we can find a solution, we can solve a system of equations using methods such as Substitution, Elimination, or Graphing and get the required price per unit for the perfume bottle and for the wristband. I will discuss those methods in later posts.
The answer is price per unit of perfume bottle is $4 and price per unit for wristband is $3.
Cheers! ❤️
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