Now that you know what a factor is, we can use that knowledge to solve more interesting equations than x + 7 = 10. But first, another maths shortcut and a property that is used all the time in algebra.
Remember that multiplication is implied with two sets of brackets next to each other:
(2)(-7) = -14
Well the same thing applies when there are unknowns:
3 × x = 3x, a × b = ab
Also, notice that when you see just x, you can think of that as 1x where the “1” is left off as anything multiplied by “1” is the same anything. Now let’s look at one of the most used properties in maths: the distributive property.
If you had an expression like 3(2 + 4), which means 3 × (2 + 4), you could first add the numbers in the brackets then multiply: 3(2 + 4) = 3(6) = 18. But another way is to first “distribute” the 3 among the terms in the brackets:
3(2 + 4) = 3(2) + 3(4) = 6 + 12 = 18
This holds true for any numbers:
4(6 – 5) = 4(6) – 4(5) = 24 – 20 = 4
-3(5 – 3) = (-3)(5) – (-3)(3) = -15 + 9 = -6 (Remember the sign rules?)
This is called the distributive property. But it’s not that useful to use with known numbers as I would normally just do the math in the brackets first. Now let’s use this property with unknowns:
x(5 – 3) = 5x – 3x. Now you might want to write x5 – x3, but it is customary to write the known number first, then the unknown. You can do this because when you multiply numbers together, it doesn’t matter what order you do the multiplication (this is another property called the commutative property).
Now notice that if you first did the calculation in the brackets first, you would get the answer 2x. Well because of the distributive property, you can add things like 3x and 2x together to get 5x because you can un-distribute the x to get
3x + 2x = x(3 + 2) = 5x
This will work for any combination of a number multiplying an unknown. When the unknown is the same letter, or combination of letters, these are called like terms and you can add them together because of the distributive property.
14a – 11a = 3a, because 14a and -11a are like terms, however,
3x + 4a cannot be added together because the unknowns are different.
-2y + 3y = 1y = y (remember the “1” doesn’t need to be shown)
5m – 2m = 3m
102x + 33x = 135x
20x – 5x + 2y = 15x + 2y (only the like terms can be added)