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.
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.
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.
Sure! I will provide three examples for you: one with infinite solutions, one with no solutions, and one with one solution. I will also provide a real-world scenario that goes along with at least one of the systems of equations.
1. System with Infinite Solutions:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20
To check if there are infinite solutions, let's manipulate these equations:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20
If we divide Equation 2 by 2, we get:
2x + 3y = 10
As we can see, Equation 1 and Equation 2 are equivalent. They represent the same line. This means that any point (x, y) that satisfies one of the equations will automatically satisfy the other. Thus, this system has infinite solutions.
Real-World Scenario: Two friends are finding a way to split the payment of their dinner bill based on the number of items they ordered. The equations represent the total amount each friend owes based on the number of items they individually ordered. Since there are multiple ways they can split the bill evenly, the system has infinite solutions.
2. System with No Solutions:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 5
To check if there are no solutions, let's manipulate these equations:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 5
If we divide Equation 2 by 2, we get:
2x + 3y = 2.5
As we can see, Equation 1 and Equation 2 are not equivalent. They represent parallel lines that will never intersect. Since there is no point that satisfies both equations simultaneously, this system has no solutions.
Real-World Scenario: A company produces two types of products, Product A and Product B. The equations represent the labor costs per unit of these products. However, Product A requires more labor than Product B, and the total labor cost is lower than what Equation 1 represents. Thus, there is no way for the company to meet these labor cost requirements, resulting in no solutions.
3. System with One Solution:
Equation 1: 2x + 3y = 7
Equation 2: 4x + 6y = 16
Solving this system yields a unique solution.
Let's multiply Equation 1 by 2 and Equation 2 by -1, then add them together:
4x + 6y = 14
-4x - 6y = -16
-----------------
0 = -2
Here, we obtain an inconsistency, where 0 does not equal -2. Since our equations became contradictory, there is no possible point that satisfies both equations simultaneously. Therefore, this system has no solutions.
Real-World Scenario: Two entrepreneurs are trying to calculate their total revenue based on the number of products sold. Equation 1 represents the revenue from Product A, and Equation 2 represents the revenue from Product B. However, there is a mistake in the calculations, leading to contradictory revenue values for the same number of products sold. As a result, the system has no solutions.
Sure! Let's start by writing a system of linear equations with infinite solutions:
System 1:
2x + 3y = 6
4x + 6y = 12
To solve this system, we can manipulate the equations to see if they are proportional:
Equation 1 multiplied by 2: 4x + 6y = 12
As we can see, Equation 2 is a multiple of Equation 1. This means that the two equations actually represent the same line. Therefore, the system has infinitely many solutions. The equations are dependent on each other and can be satisfied by an infinite number of (x, y) pairs that lie on the line.
Now let's write a system of linear equations with no solutions:
System 2:
3x + y = 5
6x + 2y = 9
To solve this system, we can manipulate the equations to see if they are inconsistent:
Equation 2 multiplied by 2: 12x + 4y = 18
Equation 1 multiplied by 4: 12x + 4y = 20
As we can see, the resulting equations are not equivalent or proportional. They represent parallel lines that never intersect. Therefore, this system has no solutions.
Finally, let's write a system of linear equations with one solution:
System 3:
2x + y = 8
3x - y = 4
To solve this system, we can use the method of elimination:
Multiply Equation 1 by 3: 6x + 3y = 24
Multiply Equation 2 by 2: 6x - 2y = 8
Adding the two equations, we get: 6x + 3y + 6x - 2y = 24 + 8
This simplifies to: 12x + y = 32
Now, subtracting Equation 1 from the simplified equation, we get:
12x + y - (2x + y) = 32 - 8
This simplifies to: 10x = 24
Dividing both sides by 10, we find x = 2. Substituting this value back into Equation 1:
2(2) + y = 8
4 + y = 8
y = 4
Therefore, the solution to System 3 is x = 2 and y = 4.
Real-world scenario:
Consider a scenario where you are managing a lemonade stand. You know that each cup of lemonade you sell costs $2 to make and you are selling them for $5 each. Thus, the amount you make from selling cups of lemonade can be represented by the equation:
Revenue = 5x
However, you also have fixed costs of $14 per day to cover expenses such as cups, lemons, and sugar. Thus, your total cost can be represented by the equation:
Costs = 14
To determine the profit (revenue minus costs), you can subtract the cost equation from the revenue equation:
Profit = Revenue - Costs = 5x - 14
Now, let's say you want to determine how many cups of lemonade you need to sell to break even (i.e., make no profit or loss). In this case, profit would be equal to 0:
0 = 5x - 14
To solve for x, we can add 14 to both sides of the equation:
14 = 5x
Dividing both sides by 5, we find:
x = 14/5
Thus, to break even in this scenario, you would need to sell approximately 2.8 cups of lemonade.