Question

Basic Algebra (operations on variables)

Answer 1

Making the transition from non-variable operations to variable operations is difficult for many students. 3 + 4/7 is much less intimidating than 3x + (4x/7y). However, I hope to make an analogy that will make variable operations easier to understand.

Answer 2

Imagine that you have a bunch of opaque bags of marbles. You don't know how many marbles are in each bag, but you are told that each bag has the same amount of marbles.

Answer 3

We can let that amount be called "x" or some other variable. Instead of saying "each bag has 3 marbles" we say "each bag has x marbles" because we don't know the amount x represents yet.

Answer 4

Now imagine we have a scale that tells us that two bags of marbles weighs the same as 4 loose marbles (pretend the weight of the bag is negligible).

Answer 5

We can represent this as 2x = 4

since 2x means 2 (number of bags) multiplied by x (marbles per bag)

Answer 6

if we want to know the number of marbles in each bag (x) we can try to create a situation where we have

x = ____ (where some quantity goes into this blank)

where x is by itself on one side of the equation. this expression tells us that one bag of marbles weighs some amount.

Answer 7

think about what is being done to x, and do the opposite

Answer 8

since x is being multiplied by 2, we can divide both sides by 2 to get x

2x = 4

x = 2

meaning that there are 2 marbles in each bag.

Answer 9

*** important note: don't forget to keep track of your variable when you do operations.

for example

10x - 8x = 2x

if you write

10x - 8x = 2

this is wrong, because 2x is not the same as 2 by itself. 2 implies 2 marbles, 2x implies 2 bags of marbles

Answer 10

now, imagine you have two differently sized bags of marbles. let's call them x and y (so one bag contains x marbles and the other contains y marbles)

let's say you have 3 of the x bag and 2 of the y bag

so

total marbles = 3x + 2y

Answer 11

3x + 2y = 5x is wrong because this implies that y represents the same number of marbles as x

Answer 12

in addition, you cannot combine the number of marbles in a bag with loose marbles

for example, 3 x-bags of marbles + 4 loose marbles can be represented as

3x + 4

you cannot combine these to get 7x or 7, because the 3 in 3x implies 3 bags, not 3 loose marbles, so it cannot be combined with the 4 loose marbles

Answer 13

these are some of the most common mistakes I see with variable operations. I hope this tutorial helped you, and don't hesitate to ask me questions if you are confused.