This is an example homework on how to solve equation using basic algebra. Brought to us by: Calcustar.com.
Mr. Hogan hired Steve, his son to mowed their lawn. Can you compute how much Steve will earned?
We can write Steve’s Earnings as a math equation:
Steve’s Earnings = (Fixed Earnings) + (30 x Number of lawns)
Day One: Steve’s Earnings = 50 + 30 x 0 = 50 + 0 = $50. As you know $50 is fixed part and 0 is number of lawns on day one.
Day Two: Steve’s Earnings = 50 + 30 x 2 = 50 + 60 = $110. As he mowed 2 lawns on day 2, hence, 30 is multiplied by 2.
Day Three: Steve’s Earnings = 50 + 30 x 5 = 50 + 150 = $200. Because Steve mowed 5 lawns on day three, so 30 is multiplied by 5.
Day Four: Steve’s Earnings = 50 + 30 x 1 = 50 + 30 = $80. Finally, he mowed only 1 lawn on day four, hence 30 is multiplied by 1.
Analyze Steve’s earnings for all of the four days.
The earnings for all four days are not same; they are different for different days. In other words Steve’s earnings for each and every day are changing and changing quantities in mathematics are called “variables”.
Hence, we can say that Steve’s earnings can be represented by using a variable. Now, we can say that Steve’s Earnings are changing or we have another option to avoid this lengthy sentence “Steve’s earnings are changing”.
So, all the Mathematicians of the world adopt a standard to represent the variable quantities (or variables), by using letters from the alphabet. Generally the lower case letters are used to represent variables.
Hence, we can use a letter to represent the Steve’s earnings. Consider that letter is “e”. Remember that Steve’s earnings for any day is always a number in dollars and letter “e” is the common representative of earnings.
Therefore, we pick the letter “e” to represent Steve’s earning for any day he worked for his dad. Now, as you already know that Steve’s earnings (e) depend upon the number of lawns he mowed, which is again not fixed for the day. In other words, the number of lawns mowed by Steve is another variable in our example.
And we can represent it by using another lower case letter (other than “e” because two different variables need different symbols), from the alphabet. Let’s use letter “n” to represent the number of lawns mowed during a day.
Now we can write both the variables as shown below:
Steve’s earnings for a day = e
Number of lawns mowed by Steve = n
We have accomplished a great concept of algebra1, the variables. There are two variables (changing activities) in the above example. Now, recall your thinking process. There is something common happening, in terms of math operations, in the above calculations of earnings for all the four days.
To find Steve’s earnings (e), 50 is added to 30 times the number of mowed lawns by Steve.I sn’t this process is common for all of the four days? Yes, it is.
This common relationship between the earnings (e) and number of lawns mowed (n) is actually algebra1, and knowing this type of relations is, the knowledge of algebra1.
Mathematically, we can write your thinking process as given below:
Earnings for the day = 50 + 30 x Number of lawns mowed
Above is an example of algebraic relation between two variables. You know that the Steve’s earnings for the day are not fixed and are denoted by using letter “e”, also the number of lawns mowed by him are not fixed and denoted by the letter “n”.
Hence, the above algebraic relation can be rewritten using symbols (variables) for simplicity, as shown below;
e = 50 + 30 x n
Note that we can write “30 x n” as “30n” as there is no need to show multiply sign between the number and its variable as it is understood for math purposes.
e = 50 + 30n
The above is a simple algebraic expression between two variables. And both the variables are talking about an everyday life situation.
Notes about the variables used in above algebraic expression:
1. Steve’s earning for the day: Steve’s earnings are not same each day or his daily earnings are changing or unknown until he finishes his work for the day. Any unknown or changing activity is called a variable in mathematics, so Steve’s daily earnings are variable and letter “e” is used to represent it.
2. Number of lawns mowed per day: The number of lawns mowed are not same for each day. This is the second variable and we used letter “n” to denote it.Next point to note that Steve’s earnings (e) depend upon the number of lawns mowed (n). Hence, there is a variable depending on the other.
In other words, “n” is independent variable as it does not go up or down with earnings, actually it derives the earnings up or down. Earnings “e” are the dependent variable. Rewrite our algebraic expression again as follows:e = 50 + 30nIn above expression, letters “e” and “n” are the variables.
The fixed value 50 is called the constant term; remember that, constant terms are numbers without any variables.30, the multiplier of ‘n’ and is called the coefficient of “n”, the variables have numbers multiplied to them called coefficients.
Did you notice that the variable “e” does not have any number multiplied to it. In algebraic relations if a variable is written alone, it got the coefficient ONE. Therefore, “e” is actually “1e”.
Algebra is branch of mathematics which deals with changing or unknown activities (variables) in our daily life.Letters from the alphabet are used to represent variables.
Variables are actually used to represent numbers, but these numbers are unknown till the right time or certain conditions are met. A constant term in an algebraic relation is a fixed number.
As in the given example, Steve knows that he will get $60 per day he worked, which is a constant of the give expression. A coefficient is a number multiplying to the variable. In the given example 30 is getting multiplied by variable, “n”. Hence, 30 is coefficient of n.
The only letter (variable) in an algebraic expression, have got coefficient “ONE” which need not to be shown and is understood in mathematics. Hence the lessons on algebra1 for beginners.
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