Question 15: The length of the 3rd side is a-2 units. (I will answer Question 16 as a comment, by the way)
Step-by-step explanation:
Question 15:
Perimeter is the sum of all the side lengths of a shape, in this case, the shape is a triangle. Two side lengths and the perimeter is given, and you have to find the length of the 3rd side.
First, you'll need to add both of the given side lengths together. You do that by making an equation, where you can say that a question mark represents the unknown side.
2(a+3) + (3a-1)+ ? =6a+3
Then, you will need to use the distributive property for the first side. You have to distribute the 2 to the expression inside the parentheses. You need to multiply 2 by a, and by the 3. So, the expression becomes 2a+6.
Next, we'll work on the second side. We have to open up those parentheses that hold 3a-1 inside them. You can do that by distributing the plus sign in front of the parentheses to the expression inside the parentheses. You are just trying to either change or keep the sign in front of the expressions inside the parentheses.
We distribute the + to the 3a by multiplying, and it stays as 3a because the 3a is already a positive expression. If there is no minus sign in front of the 3a, that means that it is positive, and a positive sign multiplied by a positive number gives you a positive product. Now, we need to distribute the plus sign to the second term in the parentheses. Attention, the second term is a -1, not a positive 1. We can get that by changing 3a-1 to 3a+ -1. It is the same thing, it just looks a bit different. In words, the expression would be 3a plus a negative 1, instead of 3a minus 1. After that, we finally multiply the + by -1, which would be -1. That is because a positive sign multiplied by a negative sign will give you a negative product. Finally, after distributing the plus sign to the terms inside the parentheses, the expression becomes 3a-1. It stayed the same as it was, but it is still important to understand why it was not +1.
Now, it's time to re-write the whole expression.
2a+6+3a-1+?=6a+3
The next step is to combine the like terms on the left side of the equal sign. The terms that include the variable "a" come together, and the numbers (or constants) are combined. We combine by adding or subracting the terms. 2a+3a=5a. 6-1=5. This is why it was so important to open up those parentheses in the beginning and distribute, to not get the signs confused. If you skip that step, your answer could be incorrect. We leave the ? alone because there are no other terms that have a ? in them.
The new equation now looks like this: 5a+5+?=6a+3
Then, you need to simplify this equation by moving all of the terms that have an "a" in them to the same side of the equal sign, and the same thing to the constants (the numbers). We need to separate the terms from the question mark because that's what we're solving for. As we move the terms from the left side to the right, their sign changes to the opposite. So, 5a becomes -5a, and +5 becomes -5.
The new equation looks like this: ?=6a+3-5a-5
The next step is to again, combine the like terms, but on the right side of the equal sign. We leave the ? alone for now. 6a-5a=a. +3-5= -2.
The last step would be to just re-write the equation.
?=a-2 That is your answer. The length of the 3rd side is a-2 units.
Key take-aways of this problem:
- Perimeter is the sum of all of the lengths of all of the sides of a shape.
- It's important to look at the sign in front of the parentheses to know how to distribute it to the expresssion inside.
- To simplify an expression, you need to combine the like terms that are in it. The same variables go together, and the numbers do, too.
- When moving terms in an equation to the other side of the equal sign, the term's sign changes to the opposite.
