4) x = -2/3 and y = -4/3
5) x = 87/9 and y = 7/3
6) x = -3 and y = -13/4
Step-by-step explanation:
I will use a few methods to show how they work.
4: Let's use elimination.
The point is to add the two equations together so one of the variables cancel out.
2y = 4x
3y = 9x + 2
So, there is no obvious way to get any part the same, but can we divide the top equation, or the bottom equation or even both to make there be similar terms. There is! If we divide the first equation by 2 and the second equation by 3 we will have a y in both.
(2y = 4x )/2 becomes y = 2x
(3y = 9x + 2)/3 becomes y = 3x + 2/3
You can think of it like dividing both sides by 2 and 3 respectively. Let me know if that doesn't make sense. After that though now if you subtract one from the other you will eliminate y
y = 2x
y = 3x + 2/3
Let's subtract the lower one from the top one. y-y = 0, so it is eliminated, 2x-3x = -x and 0 - 2/3 = -2/3 since you can think of y = 2x as the same as y = 2x + 0
y = 2x
-(y = 3x + 2/3)
0 = -x - 2/3
From here just use algebra to solve for x. Let me know if you have trouble with it. I am just going to add x to both sides.
0 = -x - 2/3
x = -2/3
Now you can plug this into either of the other two and you will get the right answer. I will show you.
y = 2x
y = 2(-2/3)
y = -4/3
y = 3x + 2/3
y = 3(-2/3) + 2/3
y = -6/3 + 2/3
y = -4/3
Again let me know if something doesn't make sense.
5. I am going to use substitution
For substitution you wan to get one equation equal to one variable, then plug that into the other equation. That may sound confusing but hopefully seeing it will help.
x + y = 12
2x - 7y = 3
It would be easy to make the top equation be x= or y=, which is what we want. I will do y= so that makes x + y = 12 into y = 12 - x. let me know if that doesn't make sense. Now we have:
y = 12 - x
2x - 7y = 3
Now we use the first equation and make the y in the second be that. so 2x - 7y = 3 becomes 2x - 7(12 - x) = 3 because y = 12-x
Let's focus on 2x - 7(12 - x) = 3 and solve for x.
2x - 7(12 - x) = 3
2x - 84 + 7x = 3
9x - 84 = 3
9x = 87
x = 87/9
Now just like the last one you can plug x into either of the other two to find y. I am just going to do one so you can try the other.
x + y = 12
87/9 + y = 12
y = 12 - 87/9
y = 7/3
Not very nice number, but they are the answers.
You can think of the third way as graphing them and seeing where the two lines intersect. First let's get them into y=mx+b form. If there are more than two or three variables you wouldn't be able to draw them out, but the method would still work. So with this problem, or any of them, you could draw them as lines and see where they intersect.
1/2x - 2y = 5 becomes y = 1/4x - 5/2
3/4x - y = 1 becomes y = 3/4x - 1
Let me know if you don't understand that. But to find the intersection of two lines you just set them equal to each other so 1/4x - 5/2 = 3/4x - 1. Keep in mind they have to be y=, or x =, but you will usually find functions as y= Anyway, let's solve the new equation.
1/4x - 5/2 = 3/4x - 1
-3/2 = 2/4x Here multiply both sides by 4 and divide by 2
-3 = x
Now AGAIN we plug into one of the two equations to find y
1/2x - 2y = 5
1/2*-3 - 2y = 5
-3/2 - 2y = 5
-2y = 13/2
2y = -13/2
y = -13/4
So all three have the same final step of plugging in when you find the value of a different variable. I know I said this a lot, but let me know if you have any questions, I know this can be confusing. And you could have used an three methods to solve any of these problems.
