LMT - Linear Relationships Lesson

Linear Relationships

In previous math courses, you have learned to translate words to expressions and vice versa. You have learned what words indicate the operations of addition, subtraction, multiplication, and division.

Here are some examples that may help you remember some of the common translations of words to expressions. 

The sum of a number and 3	$x+3$
The product of a number and 3	$3x$
The sum of two numbers	$x+y$
Three times the sum of two numbers	$3\left(x+y\right)$
Three times a number	$3x$
Three less than a number	$x-3$
A number less 3	$x-3$
Three more than a number	$x+3$
A number, plus 3	$x+3$
The square of a number	$x^2$
The square of three times a number	$\left(3x\right)^2$
Three times the square of a number	$3x^2$
One-third of a number	$\frac{x}{3}\:or\:\frac{1}{3}x$
One less than 3 times a number	$3x-1$
Two more than 5 times a number	$5x+2\:or\:2x+5$
The square of the sum of two numbers	$\left(x+y\right)^2$
The sum of the squares of two numbers	$x^2+y^2$

 Here's an example:
A movie theater has 250 bags of popcorn when it opens for the Saturday afternoon showing. Every minute, as movie goers are entering the theater, 2 bags of popcorn are sold. Write an equation which represents the number of bags of popcorn the movie concession has after x minutes.

In this example, we start with 250 bags of popcorn. From this, we are subtracting 2 bags of popcorn per minute, or 2x, where x represents the number of minutes.

y = 250 - 2x

Let's look at one more.
At a golf course, it costs each player $25.00 per round. Each player must also pay a $10.50 cart fee.

If x represents the number of players who play one round of golf with a cart rental fee, write an equation that represents the amount of money paid to the golf course from these x golfers.

y = 35.50x

 Since each player has to pay the round fee and the cart fee, they each pay $35.50. Then, you multiply that by the number of players, and voila!

When real-world situations can be modeled with a linear equation, we can say that there is a constant rate of change. Remember when we first discussed the slope, or rate of change, or a linear function? Since it is a line, then the slope will be the same no matter what two points you are considering. Therefore, the rate of change of a linear function is constant, or never changing. Don't forget that the y-intercept represents where the function has an x equal to zero.

Let's investigate how we can model this.....
Mr. Bob tutors calculus students. He charges $50 for the first 2 hour session. The equation y = 20x + 50 can be used to find the total amount of money he charges a student. If y is the total amount he earns, and x is the number of hours after the first two, write a statement that describes the rate of change in the amount of money he earns.

The money Mr. Bob earns depends on how many hours he spends tutoring a student. The amount of money earned from that student is the y variable, and the x variable represents the length of the tutoring session. The slope, or rate of change, is 20. For every additional hour of tutoring after the first two hours, Mr. Bob's earnings increase by $20.

Watch this video for another example:

Use the graph above to answer the following questions:

Assignment

Now that you have spent some time learning about Linear Relationships, you are ready to complete your Linear Relationships Homework. Download the Linear Relationships Homework Links to an external site..

Once you have completed the Linear Relationships Homework, check your even answers and make sure you ask your teacher if you have any questions Links to an external site.. When you feel confident in your work, you'll need to take the Linear Relationships Homework Quiz. Make sure you have your Homework completed and with you when you log in to take it. This is a timed quiz and you'll only have time to enter the answers from your HW. You will not have time to work them out during the quiz.

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