write three linear equations: one with infinite answers, one with no solution, and one with one solution. write a real world solution that goes with at least one system of equations. show your work
1) Infinite solutions:
2x + 4y = 10
4x + 8y = 20
This system of equations has infinite solutions because both equations are essentially the same when doubled.
2) No solution:
3x - 6y = 12
6x - 12y = 24
This system of equations has no solution because the second equation is simply a multiple of the first equation, so the lines are parallel and never intersect.
3) One solution:
2x + 3y = 11
4x - y = 7
To find the solution for this system of equations, we can solve for y in the second equation:
y = 4x - 7
Substitute y into the first equation:
2x + 3(4x - 7) = 11
2x + 12x - 21 = 11
14x = 32
x = 32/14
x = 16/7
Now substitute x back into the second equation to find y:
4(16/7) - y = 7
64/7 - y = 7
-y = 49/7 - 64/7
y = -15/7
So the solution is x = 16/7 and y = -15/7.
Real world example:
The first system of equations with infinite solutions could represent a scenario where a company is offering a buy one get one free promotion on a product. The price for two items is $10, but each individual item is $2. Customers can choose any combination of items to get the discount.
**Note: Numbers have been chosen for simplicity and better understanding.