Sure! Here are three systems of linear equations with different solutions.
1) System with Infinite Solutions:
Equation 1: 2x + 3y = 8
Equation 2: 6x + 9y = 24
This system has infinite solutions because the second equation is obtained by multiplying both sides of the first equation by 3. Thus, these equations represent the same line, and every point on the line is a solution.
Real-world scenario: Suppose you and your friend are donating items to a charity. You have collected 8 cans of soup (represented by x) and 24 packs of noodles (represented by y). Each can of soup weighs 2 pounds, and each pack of noodles weighs 3 pounds. Your friend, on the other hand, has collected 6 cans of soup and 9 packs of noodles. Both of you donated the same amount of food, just in different proportions.
2) System with No Solutions:
Equation 1: 3x + 5y = 10
Equation 2: 6x + 10y = 7
This system has no solutions because when we try to solve it, we reach an inconsistent statement (0 = -3).
Real-world scenario: Consider a grocery store selling apples (represented by x) and oranges (represented by y). Equation 1 represents the cost of 10 items, where apples cost $3 each and oranges cost $5 each, and the total expenditure is $10. Equation 2 represents the cost of 7 items, where apples cost $6 each and oranges cost $10 each, and the total expenditure is $7. However, it is not possible for these two scenarios to exist simultaneously, as they contradict each other.
3) System with One Solution:
Equation 1: 2x + 5y = 9
Equation 2: -3x + y = 2
This system has one unique solution since the two equations represent two distinct lines that intersect at a single point.
Real-world scenario: Imagine you are hosting a birthday party and you want to buy balloons and cupcakes. Equation 1 represents the cost of 9 balloons (represented by x) and 2 cupcakes (represented by y), where each balloon costs $2 and each cupcake costs $5. Equation 2 represents a scenario where you decide to buy 2 balloons and 1 cupcake. However, the prices mentioned in both equations cannot be changed or mixed, and they meet at a specific point where you can find the exact cost of a balloon and cupcake.
Now try writing and solving your own systems of linear equations thats not too long: one with infinite solutions, one with no solutions, and one with one solution. Write a real-world scenario that goes with at least one system of equations. Show your work.
3 answers
solve 2x + 5y = 9
To solve the equation 2x + 5y = 9, we can follow these steps:
Step 1: Choose a variable to isolate. Let's isolate x.
2x + 5y = 9
Step 2: Eliminate the coefficient in front of the variable by using opposite operations. In this case, we can eliminate the 2 in front of x by subtracting 5y from both sides of the equation.
2x + 5y - 5y = 9 - 5y
This simplifies to:
2x = 9 - 5y
Step 3: Solve for x by dividing both sides of the equation by 2.
2x/2 = (9 - 5y)/2
This simplifies to:
x = (9 - 5y)/2
Therefore, the solution to the equation 2x + 5y = 9 is x = (9 - 5y)/2, where y can take any real value.
Step 1: Choose a variable to isolate. Let's isolate x.
2x + 5y = 9
Step 2: Eliminate the coefficient in front of the variable by using opposite operations. In this case, we can eliminate the 2 in front of x by subtracting 5y from both sides of the equation.
2x + 5y - 5y = 9 - 5y
This simplifies to:
2x = 9 - 5y
Step 3: Solve for x by dividing both sides of the equation by 2.
2x/2 = (9 - 5y)/2
This simplifies to:
x = (9 - 5y)/2
Therefore, the solution to the equation 2x + 5y = 9 is x = (9 - 5y)/2, where y can take any real value.