One point on both lines
Take the pair of equations x + y = 10 and x − y = 4. Each one draws a straight line, and the pair of values that makes both true is the point where the two lines cross.
The lines cross at one point, where x = 7 and y = 3.
Add the two equations
At the solution, both equations are true at the same time. So x + y is the same amount as 10, and x − y is the same amount as 4. Adding equal amounts to equal amounts gives equal totals, so we may add the left sides together and the right sides together: (x + y) + (x − y) = 10 + 4.
On the left, the +y and the −y cancel, because y − y = 0. That leaves x + x = 2x, so 2x = 14, and x = 7. One letter has been removed, and removing a letter this way is called elimination.
Now find y. Put x = 7 into the first equation: 7 + y = 10, so y = 3. Check it in the other equation: 7 − 3 = 4. The solution is x = 7 and y = 3.
Subtracting works too
Subtracting the equations removes x instead: (x + y) − (x − y) = 10 − 4. Take care with the signs. Subtracting −y is the same as adding y, so the left side is x + y − x + y = 2y. So 2y = 6 and y = 3, as before.
Add or subtract?
Look at a letter that has the same number in front of it in both equations. If its signs are different, like +y and −y, add the equations. If its signs are the same, subtract them.
Take 3x + 2y = 16 and x + 2y = 8. Both have +2y, so subtract the second equation from the first: 3x − x = 2x and 16 − 8 = 8, so 2x = 8 and x = 4. Put x = 4 into x + 2y = 8: 4 + 2y = 8, so 2y = 4 and y = 2. Check in the first equation: 3 × 4 + 2 × 2 = 12 + 4 = 16.
The usual mistakes
Adding when the signs match. Adding 3x + 2y = 16 and x + 2y = 8 gives 4x + 4y = 24, which still has both letters in it. Nothing was eliminated, so subtract instead.
Subtracting only the left sides. The right sides are subtracted too: 16 − 8 = 8, not 16.
Stopping at x = 7. The question asks for both letters, so put x back into one equation to find y, and check the pair in the other equation.