Classify each system of equations as having a single solution, no solution, or infinite solutions.

To find what the answer is for this problem, we need to find out whether each of them have infinite, no, or single solutions. We can do this individually.

Starting with the first one, we need to convert both of the equations into slope-intercept form. y = -2x + 5 is already in that form, now we just need to do it to 4x + 2y = 10.

2y = -4x + 10
y = -2x +5

Since both equations give the same line, the first one has infinite solutions.

 Now onto the second one. Once again, the first step is to convert both of the equations into slope-intercept form. 

x = 26 - 3y becomes
3y = -x + 26
y = -1/3x + 26/3 

2x + 6y = 22 becomes
6y = -2x + 22
y = -1/3 x + 22/6

Since the slopes of these two lines are the same, that means that they are parallel, meaning that this one has no solutions. 

Now the third one. We do the same steps. 

5x + 4y = 6 becomes
4y = -5x + 6
y = -5/4x + 1.5
 
10x - 2y = 7 becomes
2y = 10x - 7
y = 5x - 3.5

Since these two equations are completely different, that means that this system has one solution.

Now the fourth one. We do the same steps again. 

x + 2y = 3 becomes
2y = -x + 3
y = -0.5x + 1.5

4x + 8y = 15 becomes
8y = -4x + 15
y = -1/2x + 15/8

Once again, since these two lines have the same slopes, that means that they are parallel, meaning that this one has no solutions. 

Now the fifth one. 

3x + 4y = 17 becomes
4y = -3x + 17
y = -3/4x + 17/4

-6x = 10y - 39 becomes
10y = -6x + 39
y = -3/5x + 3.9

Since these equations are completely different, there is a single solution. 

Last one!

x + 5y = 24 becomes
5y = -x + 24
y = -1/5x + 24/5

5x = 12 - y becomes
y = -5x +12

Since these equations are completely different, this system has a single solution. 

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