System with infinite solutions:
Let's consider the situation at a carnival where the ticket prices for different rides are the same.
Let x represent the cost of the Ferris wheel, and y represent the cost of the roller coaster.
Equation 1: x + y = 10
Equation 2: 2x + 2y = 20
To see that there are infinite solutions in this system, we can divide the second equation by 2:
Equation 2: x + y = 10
We can see that both equations are the same, meaning any value of x and y that satisfies one equation will satisfy the other.
Thus, there are infinite solutions for this system.
System with no solution:
Now let's consider a scenario with two different production facilities each trying to meet certain production quotas. Facility A needs to produce x items, and Facility B needs to produce y items.
Equation 1: 3x + 4y = 50
Equation 2: 3x + 4y = 60
There cannot be a solution to this system because both facilities cannot meet the different quotas at the same time.
System with one solution:
Let's consider a real-world scenario where a farmer wants to plant two different types of crops (x and y) and needs to meet specific land requirements.
Equation 1: 2x + 3y = 100 (represents total acreage)
Equation 2: x - y = 10 (represents difference in acreage between crops x and y)
We can solve for x and y using various methods, such as substitution or elimination, but we'll use elimination here:
First, we'll multiply the second equation by 3, so the y coefficients match:
Equation 2 (modified): 3x - 3y = 30
Now we can add both equations:
Equation 1: 2x + 3y = 100
Equation 2 (modified): 3x - 3y = 30
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5x = 130
Now we can solve for x:
x = 130 / 5
x = 26
Now we can substitute x back into either equation to find y. We'll use Equation 2:
26 - y = 10
y = 26 - 10
y = 16
So the unique solution to this system is x = 26 and y = 16, meaning the farmer should plant 26 acres of crop x and 16 acres of crop y.
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.
2 answers
little bit shorter