Example:

Solve the 2 simultaneous equations:

5X + 4Y = 10 (1)

3X + 9Y = 16 (2)

Step 1: Multiply (1) by a factor of 3: 15X + 12Y = 30 (3)

Step 2: Multiply (2) by a factor of 5: 15X + 45Y = 80 (4)

Step 3: Subtract (3) from (4): (15X + 45Y) - (15X + 12Y) = 80 - 30

=> (15X - 15X) + (45Y - 12Y) = 50

=> 33Y = 50

=> Y = 50/33

Step 4: Substitute this value of Y into (1) to get: 5X + 4(50/33) = 10

=> 5X = 10 - (200/33)

=> 5X = 130/33

=> X = 26/33

So our solutions are: X = 26/33 and Y = 50/33

Explanation:

Steps 1 and 2:

We are looking to eliminate one of the variables X or Y in order to solve a simpler equation.

In our example we decided to eliminate the X's of both equations but we could have easily eliminated Y instead

This is done by finding the Least Common Multiple of the coefficients* of X in both equations.

In our example, the Lowest Common Multiple is 15 so we multiply our equations accordingly.

Step 3:

Now that our X coefficients are the same for both equations, we can now subtract one from the other resulting in a linear equation with only one variable: Y.

This equation is very easy to solve and will give our value of Y.

Step 4:

Now we need to find our value of X. By "plugging in" our value of Y into either (1) or (2), we can easily form a linear equation involving just X, which can also be solved very easily.

*the coefficient of something is the number before the variable ie. the coefficient of 5X is 5.

Note: The solutions you get might look "ugly" like in this example so it is worth checking by plugging in both values of X and Y into (1) to see if the solutions actually work.

Further Questions:

1. 7X + 5Y = 14
3X + 15Y = 21

2. 12X - 8Y = 19
4X + 12Y = 13

3. 30X + 14Y = -15
-25X - 39Y = 90