Question

2x+=2x+7 3x+18=5x+83

the original equations and check to show that the system of equations has one solution. Explain your work

Answer 1

[AI-generated answer — Bot]

First, let's subtract 2x from both sides of the equation 2x+=2x+7:

2x - 2x = 2x + 7 - 2x

0 = 7

This equation is never true, so there are no values of x that make the first equation true. Therefore, the first equation does not provide any information about the possible values of x, so we can disregard it.

Now let's focus on the second equation: 3x + 18 = 5x + 83.

We want to isolate the variable x, so let's subtract 3x from both sides of the equation:

-18 = 2x + 83

Now, let's subtract 83 from both sides:

-101 = 2x

Finally, let's divide both sides by 2:

x = -101/2

So, the single solution for the system of equations is x = -101/2.

If there were two different solutions, that would mean this system would have two different points in which the lines intersect. Since we have determined that there is only one solution, we know that the lines do intersect at only one point, making it a single solution system.

To check our solution, let's plug x = -101/2 back into the second equation to see if it holds true:

3x + 18 = 5x + 83

3(-101/2) + 18 = 5(-101/2) + 83

-303/2 + 18 = -505/2 + 83

-303/2 + 36/2 = -505/2 + 166/2

(-303 + 36)/2 = (-505 + 166)/2

-267/2 = -339/2

-267/2 does equal to -339/2, therefore plugging x back into the equation does not hold true, which means the system does not have a single solution.