"The Rules"
In elementary school and middle school students learn "The Order of Operations," which are rules about what we must do first, second, third, etc. The Order of Operations exist so that we are all able to "read" and interpret a mathematical sentence in the same way. If we did not have these rules, it would be difficult for us to communicate or arrive at the same answer. The Order of Operations help us to speak the same language and interpret "expressions" identically.
The Order of Operations are as follows:
The Order of Operations are as follows:
- Grouping Symbols like (parenthesis) or [brackets]
- Exponents
- Multiplication OR Division
- Addition OR Subtraction
Working Backwards
When we solve equations we must do the Order of Operations in reverse!
What do you mean, "work backwards?"
Well, let's go forward first. When we are solving equations, we are looking for the value of a variable. Our variable is a mystery number, and it is our job to figure out what it is.
Let me give you a preview. Say that our mysterious number is 3. If I have an expression,
5(3) - 7
the first thing I do is multiply my number by 5. The second thing I do is subtract 7. After I do these two steps, I get 8.
There's good news: an equation with only 1 variable tells us ALL the information we need to figure out what our mystery number is.
Let's turn the expression from above into an equation:
5(3) - 7 = 8
Following the order of operations, just as I did before, I would multiply my 3 by 5 and then subtract 7. The equation sign tells me that when I follow the order of operations, I will get 8 as a result.
I know that my equation is "true" because when I follow the Order of Operations, both sides of my equation equal 8. The statement, "8 = 8" is true, so my equation is giving me correct information.
Let's take that same equation, but this time I am going to replace my 3 with an x, and we will pretend we do not know that x = 3.
5x - 7 = 8
What would be the first thing I would do to the expression on the right side?
What would be the second thing I would do to the expression on the right side?
To solve an equation we need to work backwards. We have to do the Order of Operations in reverse and "undo" all of our moves.
Let me give you a preview. Say that our mysterious number is 3. If I have an expression,
5(3) - 7
the first thing I do is multiply my number by 5. The second thing I do is subtract 7. After I do these two steps, I get 8.
There's good news: an equation with only 1 variable tells us ALL the information we need to figure out what our mystery number is.
Let's turn the expression from above into an equation:
5(3) - 7 = 8
Following the order of operations, just as I did before, I would multiply my 3 by 5 and then subtract 7. The equation sign tells me that when I follow the order of operations, I will get 8 as a result.
I know that my equation is "true" because when I follow the Order of Operations, both sides of my equation equal 8. The statement, "8 = 8" is true, so my equation is giving me correct information.
Let's take that same equation, but this time I am going to replace my 3 with an x, and we will pretend we do not know that x = 3.
5x - 7 = 8
What would be the first thing I would do to the expression on the right side?
- If you said, "Multiply x times 5," you are correct!
What would be the second thing I would do to the expression on the right side?
- We would subtract seven.
To solve an equation we need to work backwards. We have to do the Order of Operations in reverse and "undo" all of our moves.
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The Opposite of...
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Is...
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If the last thing I did to one side of my equation was subtract, to "undo" the subtracting, I must add. I also need to worry about keeping both sides of the equation equal to the same amount, therefore, I will follow the rule, "What I do to one side, I must do to the other." Let's see this in practice in the equation we have been discussing above.
5x - 7 = 8
+ 7 +7
5x + 0 = 15
Since I needed to "undo" subtracting 7, I added 7 to both sides, and I added the 7 to the same type of term (numbers). After I do this I am left with...
5x = 15
At this point, I know I multiply my mystery number by 5, and after I do, I get 15. So, the last thing I did to my variable was to multiply it by 5. To "undo" this I must divide by 5. Again, what I do to one side, I must do to the other, in order to maintain the balance of the equation.
5x = 15
5 5
When I divide both sides by 5, I am undoing the multiplication. You will notice that I end up with a fraction: 5/5. 5x divided by 5 is the same as multiplying 5/5 by x. This is a fraction that reduces to 1/1 or 1, and 1 multiplied by x gives you 1x. This is how I am able to get one x by itself, and arrive at my solution, x = 3.
5x - 7 = 8
+ 7 +7
5x + 0 = 15
Since I needed to "undo" subtracting 7, I added 7 to both sides, and I added the 7 to the same type of term (numbers). After I do this I am left with...
5x = 15
At this point, I know I multiply my mystery number by 5, and after I do, I get 15. So, the last thing I did to my variable was to multiply it by 5. To "undo" this I must divide by 5. Again, what I do to one side, I must do to the other, in order to maintain the balance of the equation.
5x = 15
5 5
When I divide both sides by 5, I am undoing the multiplication. You will notice that I end up with a fraction: 5/5. 5x divided by 5 is the same as multiplying 5/5 by x. This is a fraction that reduces to 1/1 or 1, and 1 multiplied by x gives you 1x. This is how I am able to get one x by itself, and arrive at my solution, x = 3.