2x+=2x+7 3x+18=5x+83
Question 4: One Solution
few points into the original equations and check to show that the system of equations has one solution. Explain your work
To show that the system of equations has one solution, we first need to solve the given system of equations. We can use either the substitution or the elimination method. In this case, I will use the elimination method.
The given system of equations is:
1) 2x + y = 2x + 7
2) 3x + y = 5x + 83
Step 1: Subtract equation 1 from equation 2 to eliminate y variable:
(3x + y) - (2x + y) = (5x + 83) - (2x + 7)
3x - 2x = 5x - 2x + 83 - 7
x = 3x + 76
Step 2: Solve for x:
x = 3x + 76
-2x = 76
x = -38
Step 3: Solve for y using either equation 1 or 2. I will use equation 1:
2x + y = 2x + 7
2(-38) + y = -76 + 7
y = -69
The solution to the system of equations is x = -38 and y = -69. To verify that this is the only solution, we can plug these values into both equations:
Equation 1:
2x + y = 2x + 7
2(-38) - 69 = -76 - 69
-145 = -145 (This equation holds true)
Equation 2:
3x + y = 5x + 83
3(-38) - 69 = 5(-38) + 83
-114 - 69 = -190 + 83
-183 = -183 (This equation also holds true)
Since these values satisfy both equations in the system, we can conclude that the given system of equations has only one solution: x = -38 and y = -69.