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 make the senario about a arcade
Infinite solutions:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20
Solution:
Dividing Equation 1 by 2 yields: x + 1.5y = 5
Since Equation 2 is twice Equation 1, they are equivalent. Every point that satisfies Equation 1 also satisfies Equation 2. Therefore, there are infinite solutions.
No solution:
Equation 1: 2x + 3y = 10
Equation 2: 2x + 3y = 15
Solution:
Since the coefficients on both equations are the same, there is no way both equations can be true at the same time. Therefore, there are no solutions.
One Solution:
Equation 1: 3x + 2y = 15
Equation 2: x + y = 7
Solution:
Using the substitution method, solve Equation 2 for y: y = 7 - x
Substitute y in Equation 1: 3x + 2(7 - x) = 15
Expand and simplify: 3x + 14 - 2x = 15, x = 1
Substitute x back into Equation 2: y = 7 - 1, y = 6
Therefore, the solution to the system of equations is x = 1, y = 6.
Real-world scenario: In an arcade, there are two types of games: a shooting game and a racing game. The shooting game costs $3 per play, while the racing game costs $2 per play. If a group of friends spends a total of $15 and plays a total of 7 games, how many times did they play each game?
Equation 1: 3x + 2y = 15 (for the total amount spent)
Equation 2: x + y = 7 (for the total number of games played)
Using the substitution method:
x = 7 - y
Substitute x into Equation 1: 3(7 - y) + 2y = 15
Expand and simplify: 21 - 3y + 2y = 15, y = 6
Substitute y back into Equation 2: x = 7 - 6, x = 1
Therefore, the group of friends played the shooting game 1 time and the racing game 6 times.
can you rewrite it so that the words are not so big
Sure!
Infinite solutions:
- 2x + 3y = 10
- 4x + 6y = 20
No solution:
- 2x + 3y = 10
- 2x + 3y = 15
One solution:
- 3x + 2y = 15
- x + y = 7
Real-world scenario:
At an arcade, there are shooting and racing games. The shooting game costs $3 per play, the racing game costs $2 per play. A group of friends spends a total of $15 and plays 7 games. How many times did they play each game?
Solution:
- 3x + 2y = 15 (total amount spent)
- x + y = 7 (total number of games)
Using substitution method:
- x = 7 - y
- 3(7 - y) + 2y = 15
- 21 - 3y + 2y = 15
- y = 6
- x = 7 - 6, x = 1
The group played the shooting game 1 time and the racing game 6 times.