Hey there!
Given the following system:
4x + y = 8
-3x - y = 0
We do not need to change the equations at all, because the "y"s will already cancel eachother out. When we add the equations together, we get
x = 8
Now, we have to solve for y: We can do so by plugging in the value of x in to one of the equations:
4(8) + y = 8
Now we solve:
4(8) + y = 8
32 + y = 8
y = -24
The solution to the first given system is (8, -24)
Now, we can check by plugging the numbers into both equations, and seeing if they are true:
4(8) + (-24) = 8
32 + (-24) = 8
8 = 8 ~ TRUE
Now, we check the second equation:
-3(8) - (-24) = 0
-24 - (-24) = 0
-24 + 24 = 0
0 = 0 ~ TRUE
Both equations are true, which means that the correct solution to the first given system is (8, -24)
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For the second given system, we are going to have to change one of the equations because, when added, the "x" or "y" values will not cancel out.
3x + 5y = 54
6x + 4y = 72
We can divide the second equation by -2 so the "x" values will cancel out:
-3x - 2y = -36
Now, we add the equations:
3y = 18
And solve for "y":
3y = 18
y = 6
Now, we plug the "y" value into one of the equations and solve for "x":
3x + 5(6) = 54
3x + 30 = 54
3x = 24
x = 8
The solution to the second given system is (8, 6)
Now, we can check by plugging the "x" and "y" values into both equations:
3(8) + 5(6) = 54
24 + 36 = 54
54 = 54 ~ TRUE
Now, we check the second equation:
6(8) + 4(6) = 72
48 + 24 = 72
72 = 72 ~ TRUE
Both equations are true, which means that the correct solution to the second given system is (8, 6)
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Hope this helps! Have a great day, and let me know if you need more help or have questions!