Sometimes equations have repeating groups. Let's see how we can use these groups to make solving them easier. Here we have a balance scale. On the left side, we have two C circles. We also have two weights marked three, which add up to six. On the right, the two four weights give a total of eight. So the equation is 2 C + 6 = 8.
Let's look at the left side again.
Instead of seeing separate C's and threes, we can pair them up. We can form two identical groups where each group contains one C and one three. We put these groups into boxes. How many C +3 groups are there? There are two. Let's use this insight to rewrite our equation. We can represent the left side as having two of these C +3 groups. The right side remains the same with a total weight of eight. This gives us 2 * the quantity C + 3 = 8.
In algebra, we use parentheses to create these groups. An expression like 2 * C + 3 is a compact way of saying two groups and each group contains C + 3.
Let's apply this to another equation. On this scale, the left side has three weights, each marked seven for a total of 21. On the right side, we see three identical boxes. Inside each box is an unknown t plus a weight of 1. So, our equation is 21 = three groups of t + 1.
Now that we have our grouped equation 21 = 3 * t + 1, we can simplify it. Since the scale is balanced with three groups on each side, we can look at just one group from each side. One group on the left is a weight of 7. One group on the right is t + 1. This gives us the simpler equation 7 = t + 1.
This simplified equation is much easier to solve. If 7 = t + 1, we subtract 1 from both sides and find t = 6.
Let's recap. We started with a more complex equation. 21 = 3 * t + 1. By recognizing that it was made of three equal groups, we simplified it and found the solution. Noticing groups is a powerful strategy for solving equations.
