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Equations and Inequalities
Adding and Subtracting Constants
Sample Problem
Consider the equation x = 5. This equation claims that x is equal to 5.
We can create equivalent equations to x = 5 by adding or subtracting the same amount of weight to or from each side of the scale. If we add a hacky sack with weight 1 to each pan, the scale must still balance. (Yes, those are hacky sacks. Just go with it.) It now represents the equation x + 1 = 6...although it's much more difficult to play hacky sack this way.
If we add another hacky sack with weight 1 to each pan, the scale must still balance, now representing the equation x + 2 = 7. Where are we getting all these things?
If we remove two hacky sacks from each side, we are back where we started, representing the equation x = 5. Plus, we get our hacky sacks back. We were starting to worry.
Now we know how to solve equations such as x + 2 = 7. We want to isolate x on one side of the equation, so we take 2 from each side and see the scale balance at x = 5. The solution to the equation is 5.
Although the hacky sack analogy breaks down a bit with negative numbers, unless you have one of those newfangled "negative space hacky sacks," the idea is the same. We can add or subtract any number we like, as long as we add or subtract it from both sides of the equation. Otherwise, we are unbalancing the scale. The goal is to keep the scale balanced while getting the variable all by itself on one side of the equation.
Sample Problems
Solve each equation
- x - 3 = -1
Add 3 to each side of the equation to find that x = 2.
- x + 5 = -2
Subtract 5 from each side of the equation to find that x = -7.
One way to keep track of what we are doing is to write the operation we are performing under each side of the equation. Doctors will often do this sort of thing when they are performing operations: