Solving equations using the balancing method

An equation is an equality and simple equations are easily solved by the index finger method. But equations will soon become harder; like this one: 4x-1=2x+9. Here, x is on both sides and it's no longer helpful to cover one part of the equation with your index finger to see the answer. But we can break down the solution into several easy steps and eventually isolate the unknown variable on the left side. In order to do that we use the balance method.

Rule number one for the balance method is what you do on one side, you also have to do on the other side. So we bring out the balance scales again. On the left pan we have four secret boxes and minus one kilogram. The helium balloon represents minus one. On the right pan we have two secret boxes and nine kilograms.

Remember the rule. What we do on one side, we also have to do on the other side. So, let's go. We want to have x alone on one side. So, we remove two exes from the right side.

And then we have to do the same on the left side. Remove two exes. Write a new line in the equation where we add minus 2x to both sides. Now we can simplify. The exes disappeared from the right side as we wanted.

And now it's time to remove everything that is not x from the left side as well. To get rid of -1, we add one. And what we do on the left side, we also have to do on the right side. So we write +1 on each side and simplify. Do you see what happens?

For every step we take, we do the same thing with both the right and left sides. And get a new equation that's a bit simpler than the last one. We now have one that is so simple that you can solve it with the index finger method. But let's continue in this way. To show how this works.

To turn 2x into a single x on the left side, you can try to add -x on both sides. You now have a single x on the left. But you also have an x on the right again. So, this was a wrong step. Let's erase it.

Do this instead. To turn 2x into 1x, divide by 2, and what you do one side, you do on the other side as well. So you divide what you have on the left pan by 2 as well because if the whole left pan weighs as much as the whole right pan, then half of the left one weighs as much as half of the right one. We write the same thing in the equation and divide it by 2 on both sides. Then you only have to simplify to get the answer, x = 5.

Plug the solution into the original equation. You have it here. Pause the video and verify that your solution is correct. Can you handle another example? Pause the video and rest your brain for a while if you want to because now we'll do it a bit faster.

New equation. On the left side you have one quarter of y + 2, and on the right side 5 - one quarter of y. Start by getting all terms with the unknown variable on the same side. Add a quarter of y to both sides. Simplify.

We now have two quarters of y on the left and no y at all on the right. Good. To leave the y's alone on the left side, add minus 2 to both sides and simplify. To turn 1/2 of y into 1y, you multiply the whole left side by 2 and do the same with the right side. Then you only have to simplify, y = 6.

Here is the original equation again. Pause the video, plug in 6 instead of each y and see for yourself that we got it right.