Now try writing and solving your own systems of linear equations: one with infinite
solutions, one with no solution, and one with one solution. Write a real-world scenario
that goes with at least one system of equations. Show your work.
1) System with infinite solutions:
x + y = 5
2x + 2y = 10
This system has infinite solutions because the second equation is just a multiple of the first equation. They represent the same line. Any point on the line x + y = 5 is a solution.
2) System with no solution:
x + y = 5
2x + 2y = 12
This system has no solution because the two equations represent parallel lines. They will never intersect and therefore have no common solution.
3) System with one solution:
2x + y = 8
3x - y = 4
Let's say you have a fruit stand selling apples and bananas. You sell 2 apples and 1 banana for $8, and 3 apples and 1 banana for $4. You want to find the cost of each fruit.
To solve this system, we can eliminate y by adding the two equations:
2x + y = 8
3x - y = 4
-----------
5x = 12
x = 2.4
Substitute x back into the first equation to find y:
2(2.4) + y = 8
4.8 + y = 8
y = 3.2
So the cost of an apple is $2.40 and the cost of a banana is $3.20.