I'm kinda stuck on this problem,

No Solution:

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One Solution:

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Infinite:

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Explanation:

I will show one of each as an example for you to use. Here is the first one, it has one solution because it has a value for both x and y.

Let's solve your system by substitution.

−8x−10y=24;6x+5y=2

Solve−8x−10y=24 for x:

−8x−10y=24

−8x−10y+10y=24+10y(Add 10y to both sides)

−8x=10y+24

−8x−8=10y+24−8(Divide both sides by -8)

x=−54y−3

Substitute−54y−3 for x in6x+5y=2:

6x+5y=2

6(−54y−3)+5y=2

−52y−18=2(Simplify both sides of the equation)

−52y−18+18=2+18(Add 18 to both sides)

−52y=20

−52y−52=20−52(Divide both sides by (-5)/2)

y=−8

Substitute−8 for y in x=−54y−3:

x=−54y−3

x=−54(−8)−3

x=7(Simplify both sides of the equation)

x=7 and y=−8

Next, a no solution problem. We will use the second system of equations.

Rewrite equations:

−x+y=3;−3x+3y=4

Solve−x+y=3 for y:

−x+y=3

−x+y+x=3+x(Add x to both sides)

y=x+3

Substitute x+3 for y in−3x+3y=4:

−3x+3y=4

−3x+3(x+3)=4

9=4(Simplify both sides of the equation)

9+−9=4+−9(Add -9 to both sides)

0=−5

This is not possible. Thus, there is no solution.

Lastly, a infinite solution problem. We will use the last system.

Solve 2x+8y=6 for x:

2x+8y=6

2x+8y+−8y=6+−8y(Add -8y to both sides)

2x=−8y+6

2x2=−8y+62(Divide both sides by 2)

x=−4y+3

Substitute −4y+3 for x in−5x−20y=−15:

−5x−20y=−15

−5(−4y+3)−20y=−15

−15=−15(Simplify both sides of the equation)

−15+15=−15+15(Add 15 to both sides)

0=0

It is equal, meaning that there are infinite solutions.