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 solve the system of equations step by step.
1. First equation: ( y = 3x + 10 )
2. Second equation: ( 2y = 6x – 4 )
To analyze the second equation, we can divide everything by 2 to simplify it:
– ( y = 3x – 2 )
Now we have two equations to look at:
– ( y = 3x + 10 )
– ( y = 3x – 2 )
Next, let’s set them equal to find the solution:
[ 3x + 10 = 3x – 2 ]
Subtract ( 3x ) from both sides:
[ 10 = -2 ]
This is a contradiction, which means there are no values of ( x ) that make both equations true at the same time.
### Now, let’s fill in the blanks:
Mia
Mia correctly graphs the two lines and sees that they have no solution(s).
Roberto
Roberto correctly sees that ( y ) cannot 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 never be equal, the system has