Writing Linear Equations
Now try writing and solving your own systems of linear equations :one with infinite solutions one with no solution and one solution , write a real-world scenario that goes with at least one system of equations .Show your work
System 1: Infinite Solutions
Equation 1: 2x + 3y = 6
Equation 2: 4x + 6y = 12
To determine if this system of equations has infinite solutions, we need to check if both equations represent the same line.
Dividing the Equation 2 by 2, we get:
2x + 3y = 6
Both equations represent the same line (they are proportional), so this system has infinite solutions. In other words, any values of x and y that satisfy one equation will also satisfy the other.
Real-world scenario: A bakery sells muffins (x) and cupcakes (y). The total revenue from selling muffins and cupcakes in a day is $6. The price of each muffin is $2, and the price of each cupcake is $3. One day, they sold 4 muffins and 6 cupcakes. To represent this scenario, the system of equations can be written as follows:
Equation 1: 2x + 3y = 6 (total revenue)
Equation 2: x = 4 (number of muffins sold)
Equation 3: y = 6 (number of cupcakes sold)
System 2: No Solution
Equation 1: 3x + 2y = 5
Equation 2: 6x + 4y = 10
To determine if this system of equations has no solution, we can check if the slopes of the two lines are equal but the y-intercepts are different. In this case, if we divide Equation 2 by 2, we get:
3x + 2y = 5
Now both equations represent the same line (they are proportional), so they have infinitely many solutions.
Real-world scenario: A car rental company charges $5 per hour (x) plus $10 per day (y) for renting a car. A family rented a car for 3 hours and 2 days. We want to find the total cost of their rental:
Equation 1: 3x + 2y = 5 (total cost of rental)
Equation 2: x = 3 (hours of rental)
Equation 3: y = 2 (days of rental)
System 3: One Solution
Equation 1: 2x + 3y = 7
Equation 2: 4x + 6y = 14
To determine if this system of equations has one solution, we need to check if the two lines intersect at a single point. If we divide Equation 2 by 2, we get:
2x + 3y = 7
We can see that these equations are not proportional and represent different lines, so they will intersect at one point.
Real-world scenario: A restaurant offers two meal options: Option A (x) includes 2 entrees and 3 sides, and costs $7, while Option B (y) includes 4 entrees and 6 sides, and costs $14. We want to find out how much each entree and side costs:
Equation 1: 2x + 3y = 7 (cost of Option A)
Equation 2: 4x + 6y = 14 (cost of Option B)