Equations with several variables are called literal equations. The quick answer is, x is a place-holder. It's a blank slot, in which we can place a number, and then we get either a numerical expression or a numerical sentence which may be true or false. To go deeper into this, we need to recognize that letters in algebra may be used in two ways: as constant symbols or as variable symbols. A letter may stand for a number. Some letters stand for particular number. For example e by convention stands for a number near 2.
Sometimes, we use a letter in the context of a particular problem to stand for a specific number. If Ann's age is one quarter Bill's today and will be one third of Bill's five years from now, then how old is Ann? Let a stand for Ann's age today and let b stand for Bill's age today. In the example, from the moment we introduced the symbol a, it stood for the number However, we did not know that until we had done a little thinking. In this case, the symbol a is NOT a variable, it is a name for The symbol a, in this context, is called a constant symbol.
The example shows that we may have a constant symbol whose value we do not immediately know. Let c stand for my father's age on January 1, So you can probably do these in your head, think about what number x might stand for. What number plus gives you the answer 8?
That's problem one. Look at problem 2. It uses the same letter but it's going to be a different number. What number take away 4 gives us the answer 10? So the trick with variables is that it's the same letter and it represents any number like it could be, sometimes x would be like a fraction, sometimes x will be a decimal but the trick is that it might be different numbers from one problem to the next. When you come across variables, it's something that's kind of new because you're going to be dealing with letters and numbers.
We start, as usual, by factoring. For each of the denominators, we find all the prime factors —i. If you are not familiar with the concept of prime factors, it may take a few minutes to get used to. Similarly, the prime factors of 30 are 2, 3, and 5. But why does this help? Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of Prime Factors of Fractions: Finding the prime factors of the denominators of two fractions enables us to find a common denominator.
The least common denominator is the smallest number that contains the overlap of both factored denominators: in this case, it must have two 2s, one 3, and one 5. This may look like a very strange way of solving problems that you have known how to solve since the third grade.
However, you should spend a few minutes carefully following the above solution. When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same. If they are the same, then simply add or subtract the numerators from each other, leaving the denominator alone.
If the two denominators are different, however, then you will need to use the above strategy of finding the least common denominator. When we add or subtract rational expressions, we will not simply be considering the prime factors of integers when looking for the least common denominator.
Rather, we will be looking for monomial and binomial factors that are common to both rational expressions. This requires factoring algebraic expressions. We begin problems of this type by factoring. Notice that we can rewrite the first denominator in terms of its factors. The denominator in the second fraction cannot be factored. The rational expressions therefore become:.
Notice the factors in the denominators. Now, as above, we need to find the smallest possible overlap including all the factors in both of these denominators. We now rewrite both fractions with the common denominator remember that if you multiply a denominator by a factor, you must also multiply the numerator of that fraction by the same factor :. Subtracting fractions is easy once you have a common denominator!
Now we can simply subtract the numerators:. Second , follow the regular procedure for fractions, which in this case involves finding a common denominator.
Privacy Policy. Skip to main content. Introduction to Equations, Inequalities, and Graphing. Search for:. Variables and Expressions. Learning Objectives Describe the uses of variables in mathematics. Expressions such as 4x 3 which consists of a coefficient times a variable raised to a power are called monomials.
A monomial is an algebraic expression that is either a numeral, a variable, or the product of numerals and variables. Monomial comes from the Greek word, monos, which means one. Real numbers such as 5 which are not multiplied by a variable are also called monomials. Monomials may also have more than one variable. In this expression both x and y are variables and 4 is their coefficient. One or more monomials can be combined by addition or subtraction to form what are called polynomials.
Polynomial comes from the Greek word, poly, which means many. A polynomial has two or more terms i. If there are only two terms in the polynomial, the polynomial is called a binomial. The coefficients of the terms are 4, -2, and 3. The degree of a term or monomial is the sum of the exponents of the variables. The degree of a polynomial is the degree of the term of highest degree. In the above example the degrees of the terms are 5, 3, and 0. The degree of the polynomial is 5. Remember that variables are items which can assume different values.
A function tries to explain one variable in terms of another. Consider the above example where the amount you choose to spend depends on your salary. Here there are two variables: your salary and the amount you spend.
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