Like all other topics in mathematics, algebra is learned from the bottom up. If a student doesn’t understand the fundamentals, he or she will have difficulty mastering more difficult concepts that follow. One of the first concepts in algebra that needs to be mastered is simplifying expressions and combining like terms.
Many times algebraic expressions are not in the simplest form. We use the properties of real numbers mentioned in a previous article in order to rewrite an expression in a less complex form. For example, suppose w e want to simplify the expression 5(20 x ). Using the associative property of multiplication, we can regroup the 5 with the 20 to get (5∙20)∙ x . Remember the order of operations and combine what is inside the parentheses to get 100x.
We can use the same property when dealing with expressions involving fractions. For example, suppose we want to simplify the expression (9/2 )x( 4/3).
Using the associative property of multiplication, we can regroup 9/2 with 4/3 to get (9/2)(4/3). Using the commutative property of multiplication, we get (9/2)(4/3) x .
From here you can multiply first or you can simplify first. I always find it easier to simplify the fractions first then multiply. The way to simplify is to look to see if the numerators divide evenly into the denominators or the denominators divide evenly into the numerators. If not, see if there are any common factors between the numerators and denominators. Remember that the numerator is the top number in a fraction and the denominator is the bottom number in a fraction. A common factor is a number that divides evenly into both numbers.
Notice that the 2 divides evenly into 4 and 3 divides evenly into 9.( 9/2)(4/3) x = (3/2)(4/1) x (when dividing the 3 into the 9)
= (3/1)(2/1) x (when dividing the 2 into the 4)
= (6/1) x
= 6x
In a problem such as this with small numbers, simplifying is easily done if you multiply first, but with larger numbers, I suggest simplifying the fractions first before multiplying.
Now let’s consider problems that combine multiplication and addition. A problem such as 10(2 + 9) can be solved in two ways. We can use the order of operations and add the 2 and 9 first to get 10(11) = 110. We can also use the distributive property. Recall the distributive property states that for real numbers a , b and c , a ( b + c ) = ab + bc .
Using the distributive property, we distribute the 10 across the 2 and the 9 to get
10(2 + 9) = 10(2) + 10(9)
= 20 + 90
= 110
We can basically think of the distributive property as the number or term outside of the parentheses being multiplied by each term inside the parenthesis and adding the products. What if there is a problem with a fraction that needs to be distributed? You might be able to simplify first before multiplying. For example, in the problem (1/4)(8x + 24), notice that the 4 divides evenly into both 8 and 24. Simplifying first eliminated the fraction completely and we are left with 2x + 6.
Before knowing how to combine like terms, we need to define that a term is. A term is a number, variable, a product of variables or a product of numbers and variables. Examples of terms are 4, 12, -15, x , y , 2 x , 5 y , 16 xy , x2 y6 z , 2/3. In an algebraic expression, terms are separated by an addition sign. (Note you often see subtraction signs in algebraic expressions, but technically it’s addition of a negative term) Terms are considered ” like terms” if they have the same variables raised to the same power.
Example: 2 x and 3 x are like terms, 5 and -8 are like terms, x6 y and 4 x6 y are like terms but 4 x and y, xy and xz, and x2y and y2x are not like terms. To combine like terms we add or subtract the terms, keeping the variables and the exponents the same.
Example: Simplify 2 x + 6 – 5 x + 16 + 4 x by combining like terms.
Notice that x, -5x and 4x are like terms, and 6 and 16 are like terms. Therefore, (2 x – 5 x + 4 x ) + (6 + 16) = x + 22.
Example: Simplify x2 y + 2 xy + 4 xy + 5 x2 y – 14.
x2 y + 5x2y + 2xy + 4xy – 14
6x2y + 6xy – 14
This short article should provide enough helpful information and examples to assist anyone who is having difficulty mastering the concepts of simplifying algebraic expressions and combining like terms.
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