Three students are working to find the solution set of this system of equations:
y = 3x + 10
2y = 6x – 4
Use the drop-down menus to complete the statements about each of their methods.
Mia
Mia correctly graphs the two lines and sees that they __ solution(s).
Roberto
Roberto correctly sees that y __ be equal to both 10 more than three times a number and 2 less than three times a number at the same time.
Delmar
Delmar correctly concludes that because 3x + 10 and 3x – 2 will __ be equal, the system has __ solution(s).
Let’s analyze the system of equations given by:
1. ( y = 3x + 10 )
2. ( 2y = 6x – 4 )
First, we’ll simplify the second equation:
Divide the entire equation (2y = 6x – 4) by 2 to find (y):
[
y = 3x – 2
]
Now we have the two equations:
1. ( y = 3x + 10 )
2. ( y = 3x – 2 )
### Now let’s address each student’s findings:
Mia:
When Mia graphs both equations, she realizes that the two lines have the same slope but different y-intercepts. This means they are parallel and will never meet. Therefore, Mia sees that they have 0 solution(s).
Roberto:
Roberto recognizes that (y) cannot be equal to both (3x + 10) and (3x – 2) at the same time since that would require the two expressions, which are parallel, to intersect. Thus, he correctly concludes that (y) cannot be equal to both at the same time.
Delmar:
Delmar notes that since (3x + 10) and (3x – 2) are linear expressions with the same slope, they will never be equal. Thus, the system has **0