Let's figure out what balances the scale. On the left side, we have two groups of five plus 3 C. We can combine the like parts from both groups. Let's start with the constant numbers. We have a five from the top group and a five from the bottom, giving us a total of 10. Now, for the variable terms, we have 3 C in the first group and another 3 C in the second for a total of 6 C. So, the expression two groups of 5 + 3 C = 10 + 6 C.
This algebraic move where you multiply the number outside the parentheses by each term inside is called distributing.
In this case, we distributed the two. As you can see, 2 * 5 gives us 10 and 2 * 3 C gives us 6 C. Let's practice this without the visual aid. We need to rewrite the expression three groups of x + 4 by distributing the three. We multiply the three by the first term inside the parenthesis, which is x. So 3 * x gives us 3 x. Next, we multiply the three by the second term, which is 4. 3 * 4 gives us 12. So, the equivalent expression is 3x + 12.
Let's look at another example, this time involving subtraction. The expression is 10 groups of 5 aus. We'll distribute the 10 to both terms inside. First, multiply 10 by the first term, 5 a. 10 * 5 a gives us 50 a. Then, we multiply 10 by the second term, minus 8, which gives us - 80. The final expression is 50 a minus 80.
The distributive property works with more than two terms. Let's look at five groups of 2 y + 8 z + 11. We distribute the five to each of the three terms.
First, 5 * 2 y = 10 y. Second, 5 * 8 z = 40 z. And finally, 5 * 11 = 55. We then combine these new terms using the original plus signs. Our final expression is 10 y + 40 z + 55.
We've just seen how we can rewrite expressions by distributing. This is a key skill for simplifying problems and making complex equations easier to solve.
