A balanced scale means both sides weigh the same. On the left, there are two squares. On the right, there is one square and a weight of eight. If we remove one square from each side, the scale stays balanced. Now, we're left with one square on the left and an eight on the right. That tells us one square weighs eight. Now, look at this. The left side has three circles and a one.
The right side has two circles and a three. Let's simplify by removing what's the same from both sides. We can take away two circles from each side. That leaves one circle and a one on the left equal to a three on the right. Now let's subtract the one from both sides. That leaves us with one circle equal to two.
So one circle weighs two.
This is the key idea. When a scale is balanced, you can add or remove the same weight from both sides and it'll stay balanced. We can use that to figure out the value of unknown weights.
Let's try it again. The left side has five triangles and a two. The right side has three triangles and an eight. We'll start by removing three triangles from each side. That leaves two triangles and a two on the left and just an eight on the right. Now subtract two from both sides. That leaves two triangles equal to six. So one triangle must weigh three.
Now here's a problem with two different unknown shapes. First, notice there's a yellow square on each side. Since they're equal, we can remove both. That leaves a six and a circle on the left and two circles on the right. Let's remove one circle from each side. Now, we have a six on the left and one circle on the right. So, one circle weighs six.
Let's do the same thing again. Both sides have one green triangle, so we'll take those away. Now the left side shows three squares and the right side has 12 + 1 square. Let's remove one square from each side. That leaves two squares on the left and 12 on the right. So one square must weigh six.
The big takeaway is this. When both sides of a scale are balanced, you can remove equal weights from each side and the balance stays the same. This simple principle lets us isolate an unknown and figure out its value. And as you'll see, it's the foundation for solving algebraic equations.
