2x+=2x+7 3x+18=5x+83 few points into the original equations and check to show that the system of equations has one solution. Explain your work
We are given the system of equations:
(1) 2x + y = 2x + 7
(2) 3x + y = 5x + 83
First, let's solve for y in the first equation:
y = 2x + 7 - 2x
y = 7
Now, let's substitute this value for y in the second equation:
3x + 7 = 5x + 83
Now, let's solve for x:
2x = 76
x = 38
Now we have a unique solution (x, y) = (38, 7). Let's plug in these values into the original equations:
(1) 2x + y = 2x + 7
2(38) + 7 = 2(38) + 7
76 + 7 = 76 + 7
83 = 83 (true)
(2) 3x + y = 5x + 83
3(38) + 7 = 5(38) + 83
114 + 7 = 190 + 83
121 = 273 (true)
Since both original equations hold true with our solutions (x, y) = (38, 7), we can conclude that the system of equations has one unique solution:
x = 38
y = 7