2x+=2x+7 3x+18=5x+8
Question 4: One Solution
Plug a few points into the original equations and check to show that the system of
equations has one solution. Explain your work.
To check if the system of equations has one solution, we need to pick a point that satisfies both equations and explain our calculations. Initially, we can try to solve the system:
Given the system of equations:
1) 2x + y = 2x + 7
2) 3x + y = 5x + 8
We can eliminate x by subtracting the first equation (1) from the second equation (2):
(3x + y) - (2x + y) = (5x + 8) - (2x + 7)
This simplifies to:
x = 3x - 2x = 5x - 2x - 7 + 8 = 3x + 1
x = 1
Now we can plug x = 1 back into either equation (1) or (2) to find the corresponding value of y. Plugging x = 1 into equation (1):
2(1) + y = 2(1) + 7
2 + y = 2 + 7
y = 7.
So, the solution is (x, y) = (1, 7). Let's verify that this solution works for both equations:
Plugging (x, y) = (1, 7) into equation (1):
2(1) + 7 = 2(1) + 7
2 + 7 = 2 + 7
9 = 9 (True)
Plugging (x, y) = (1, 7) into equation (2):
3(1) + 7 = 5(1) + 8
3 + 7 = 5 + 8
10 = 10 (True)
Since the solution (1, 7) works for both equations, there is indeed only one solution to this system of equations.