The distributive property states that a(b+c)=ab+ac. It is used to distribute multiplication evenly to all numbers in parentheses. PEMDAS is used to solve problems with multiple operations. The distributive property can be used with variables and to simplify equations. It can also be used to work backwards to simplify solving equations.
The distributive property is expressed in mathematical terms as the following equation:
a(b + c) = ab + ac. You can read this as the sum of a(b + c) equals the sum of a times b and a times c. When you look at an equation like this, you can see that the multiplication part distributes evenly to all the numbers in the parentheses. It would be incorrect to multiply ab and add only c, or to multiply ac and add b. The distributive property reminds us that everything inside the parentheses must be multiplied by the outside number.
Students may first learn about the distributive property when they are learning about the order of operations. This is the concept that in problems where there are several mathematical operations, such as multiples, addition, subtraction, parentheses, you have to work in a certain order to get the right answer. This order is parentheses, exponents, multiplication and division. and addition and subtraction, which can be abbreviated to PEMDAS.
When you have a math problem that uses parentheses, you need to solve what’s in parentheses first before you can move on to solving other problems. If the math problem simply has known numbers, it’s easy enough to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property to 2(10) + 2(5). What becomes more complicated is when working with variables (a, b, x, y and so on) in algebra and when these variables cannot be combined together.
Consider the equation 9(10a + 2). If we don’t know what the variable a represents, we can’t add 10a + 2, but using the distributive property still allows us to simply use this expression because we know that this equation equals 9(10a) + 9(2 ). To simplify the expression we can take each part separately and multiply it by 9, and we get 90a + 18.
Another way to use the distributive property is if you want to understand similarities in an equation. In example 90a + 18, while the terms are not similar, they do have something in common. You can work backwards to drop the factor 9 and put the dissimilar terms in parentheses. So 90a + 18 can equal 9(a +2). We have removed the common factor between these terms, the common factor of 9.
Why on earth would you want to make the distributive property work backwards? Suppose we have an equation that 4a + 4= 8. Using the distributive property before we get to subtracting terms to solve for a can simplify the job. You can divide the entire equation on both sides by 4, giving us the answer a + 1 = 2. From there, it’s easy to determine that a = 1. Sometimes it makes sense to reduce dissimilar terms by their common factor to more easily solve an equation.
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