When solving a simultaneous equation using algebraic solutions how do you know when to + or - ?

Question

Heres a typical example:

3x+2y=4
5x-2y=18

And

2x+3y=2
8x+3y=17


And

3x-4y=6
5x-4y=2


I dont want to know how to solve them, all I need to know is how do you know weather to + or minus the numbers? like in the last example I know you cancell the 4y to become 0 but what about the 3x and 5x does that minus to? ( + and + = - )

Thanks for the help

Answer 1

For example, the last problem
3x-4y=6
5x-4y=2

Multiply the bottom equation (both sides) by -1 so you end up with (See note below) -1(5x-4y) = -1(2) = -5x+4y=-2

Going back to your problem with the revised bottom equation
3x-4y=6
-5x+4y=-2

Now you can add the two equations together and cancel out the 4y's at the same time.
-2x = 4
x = -2
Now you can substitute x into either equation to find y.

Note: (When you multiply both sides of the equation by the same number, you're not changing the meaning of the equation)

For example
If you have 4 = 4...
but you multiply both sides by -1, then -1(4) = -1(4)
which is equal to -4 = -4.
Even though the numbers are negative, the main things is the equality is still true, Left Side = Right Side

Answer 2

You need to add or subtract as needed to get rid of either x or y. If the coefficients of the variables you want to cancel are the same and have the same sign (both positive or both negative) then subtract. If they are the same and have different signs then add. Remeber that a - b is the same as a + (-1)(b). So subtracting one equation from another is the same as multiplying through by -1 and adding.

3x + 2y = 4
+ 5x - 2y = 18
____________
8x + 0y = 22


2x + 3y = 2
-(8x + 3y = 17) <=> -8x - 3y = -17
__________
-6x + 0y = -15

3x - 4y = 6
- (5x - 4y = 2) <=> -5x + 4y = -2
_____________
-2x + 0y = 4

Answer 3

Look at the terms that have the save numbers. Like in the first one you have +2y and -2y. If you *add* these, they will cancel to be 0. That's because they are opposites. When you add, you add everything.

3x + 2y = 4
5x - 2y = 18
-----------------
8x ....... = 22
x = 22/8
x = 11/4
x = 2 3/4

On the other hand, in the other two cases (3y and 3y, -4y and -4y), you have the same values, so you would need to *subtract* to cancel them to 0.

2x + 3y = 2
8x + 3y = 17

When you subtract, try to put the bigger terms first:

8x + 3y = 17
2x + 3y = 2
-----------------
6x ....... = 15
x = 15/6
x = 5/3
x = 1 2/3

Likewise with:
3x - 4y = 6
5x - 4y = 2

Reverse to put the bigger x terms on top:
5x - 4y = 2
3x - 4y = 6
--------------
2x ..... = -4
x = -4/2
x = -2

Notice that you need to correctly subtract 2 - 6 and get -4.

Answer 4

3x + 6y = 30 (equation a million) 1x + 6y = 20 (equation 2) Subtract equation 2 from equation a million: (3 - a million) x = 30 - 20 2x = 10 x = 10/2 = 5 replace x in equation 2 with its fee 5 + 6y = 20 6y = 20 - 5 = 15 y = 15/6 = 5/2 The pair of strategies to those simultaneous equations is (5, 5/2)