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Basic Equation Solving
In the previous set of videos, we were talking about algebraic expressions and of course, the point there was to change to equivalent expressions that would always be true for all values of x. Now we’re starting basic equation solving. And because we’ve learned a few important rules for changing algebraic expressions to equivalent expressions, this will help us in basic equation solving.
So this is a very introductory video. If you’re good at algebra, you probably don’t need to watch this video. In solving equation. The variable has either a single unknown value or a couple of unknown values. And the point is to find what the variable equals. So this is a very different game from the game of simply rearranging algebraic expressions.
Here we’re actually trying to find the value for the variable. We need to solve for x which means isolating x. That is, getting it by itself on one side of the equation. The basic mathematical constraint is that whatever we do to one side of the equation we have to do to the other. In other words we can add the same thing to both sides.
We can. We can subtract the same thing both, from both sides. We can multiply or divide. Whatever we do to one side, whatever mathematical operation, we have to do that same mathematical operation, to the other side. Mathematically, we are allowed to do many things, but strategically, there are very specific steps in solving any problem.
These steps involve undoing the order of operations. So it’s very important to have this distinction of what is mathematically allowed that’s a broad field of possibility versus strategically what are the best steps to solve the problem. So suppose we have to solve this simple equation, 3x plus 5 equals 17. Theoretically it should not be a big challenge to solve an equation like this.
But let’s talk about this. First of all, mathematically there are many things we could do. For example it would be mathematically correct, say, to add 40 to both sides of the equation. Now there’d be no reason to do that. But there’d be nothing in the laws of mathematics saying that, that was a wrong thing to do.
It would not be mathematically wrong, it would simply be strategically wrong. Just because we could do something, does not mean it is the most strategic thing to do. There are many more things that we could do than, are strategic to do. Very important to keep that in mind. So what would be strategic?
Think about the order of operations acting on x. First we multiply the x by 3, then we add 5 to the product. That’s the order we need to undo. We begin by undoing the plus 5 by subtracting 5. What we do to one side, we must do to the other. So we subtract 5 from both sides of the equation.
3x plus 5 minus 5. The 5s cancel on the left side, and on the right side, 17 minus 5 is 12. Now we have to undo the multiply by 3, so we divide by 3, and of course when we divide both sides by 3, we get x equals 4. That’s the answer. Now here’s the same equation.
Notice that we could begin by dividing both sides by 3. That would be mathematically allowed. Strategically, that would not be the smartest move. You see, if we divide it by three, every single piece, the 3x, the 5 and the 17 all would get divided by 3. And so we’d wind up with this fraction equation.
That would not be the best thing. So it’s really good to subtract before we divide. Now, why is that? Strategically, it’s always important to think about the order of operations acting on x and follow those steps backwards. So, what’s going on here?
Why backwards? Think about it this way. When you put on footwear, the order is you put on your socks, then your shoes. Of course, when you remove footwear, you don’t remove items in the same order. You have to take off the shoes first, then the socks. Undoing changes the order, both in footwear, and in many other articles of clothing.
And in mathematical operations. So just like shoes and socks, whatever goes on first comes off last. So here are a few practice problems. Pause the video and try these on your own. And here are the answers. And of course, in each case what we did was we, we added or subtracted, got rid of the constant on the same side as the x before we divided.
What if x appears on both sides of the equation? Suppose we have something like this. Here, our initial strategy is to get all the terms involving x on one side and all the constants on the other side. So, here we’re gonna begin by adding 3x to both sides, so on the, on the right side, the x’s will just cancel and on the left side we get 2x plus 3x equals 5x.
Now that we have everything on both, we have all the x’s on one side, now we can just add 7 to both sides, we get 5x equals 23, and divide, we can express the answer either as an improper fraction or a mixed numeral, or a decimal, depending on what the answer was looking for. Here are some more practice problem.
Pause the video and try these. And here the answers. Everything we have covered in this module has been linear equations. That is, equations in which the highest power of x is 1. The rules become very different when we’re no longer dealing with linear equation but everything I’ve said here is true for linear equations.
So in solving equations our goal is to find the unknown value of the variable. Mathematically, it’s, it’s always legal to do any math, arithmetic operation to both sides of the equation. Strategically, we undo the Order of Operations to isolate the variable. And if the variable appears on both sides we begin by collecting all terms with the variable, on one side.
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