Question

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.

Answer 1

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.