+2446 The table of x- and y-values definitely represent a linear equation because the increase from one point to another is constant. This is a direct indicator of a linear function.
If you cannot figure it by trial and error, then you can do this:
\(y=mx+b\)
We know that all linear functions are in this form. Let's plug in a point into this equation. I'll use the first one in the table, (2,8)
| \(y=mx+b\) | Plug in 2 for x and 8 for y. |
| \(8=2m+b\) | |
Let's do this again for a separate point. In this case, I'll use (3,11). It does not matter which point you choose. The only criteria is that it must be different from the one you previously chose.
| \(y=mx+b\) | Plug in the coordinate (3,11) into the equation. |
| \(11=3m+b\) | |
Look at this! We have a system of equations that we can solve for both m and b. I'll use substitution for this demonstration because it is easier to showcase on an online format. I'll solve for b in the first equation, 8=2m+b
| \(8=2m+b\) | Subtracting 2m from both sides is the easiest method to isolate one variable. |
| \(8-2m=b\) | |
Now, plug 8-2m in for b in the second equation.
| \(11=3m+b\) | Plug in the aforementioned value for b. |
| \(11=3m+8-2m\) | Simplify the right hand side of the equation. |
| \(11=m+8\) | Subtract 8 on both sides. |
| \(m=3\) | |
Now, substitute m=3 into either original equation. I'll do 8=2m+b
| \(8=2m+b\) | Plug in 3 for m. |
| \(8=6+b\) | Subtract 6 from both sides. |
| \(b=2\) | |
Therefore, the equation that satisfies that is \(y=3x+2\). You can confirm this by testing out each coordinate. Of course, it works.
+2446 This may help, if multiplying by 9 mentally is difficult. A lot of these mental tricks I wish I knew earlier on.
| \(15*9\) | Let's make it simpler. |
| \(15(10-1)\) | Obviously, this is the same thing. Now, distribute the 15 to both terms. |
| \(15*10-15*1\) | Now, simplify. This is much easier, don't you think? |
| \(150-15\) | And just in case borrowing is another mental headache... |
| \(150-(20-5)\) | Yet again, I have not changed the expression at all. |
| \(150-20+5\) | Simplify. |
| \(130+5\) | |
| \(135\) | |
This may look like a lot of steps, but all we are doing is making the problem easier. With practice, this will simplify multiplication with large numbers.
+9481 Hey, CPhill is just a human too! remember that!! 
( Here is the original question. )
I think the two equations should be like this....
If Alice receives n dollars from Bob, then she will have 3 times as much money as Bob.
A + n = 3 * (B - n)
If she gives n dollars to Bob, then she will have 2 times as much money as Bob.
A - n = 2 * (B + n)
So we have..
A + n = 3B - 3n → A = 3B - 4n → 3A = 9B - 12n
A - n = 2B + 2n → A = 3B + 3n → 4A = 12B + 12n Add these two to get..
7A = 17B → 7A / B = 17 → A / B = 17 / 7
