HA - Interpreting Variables and Constants in Context Lesson

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Interpreting Variables and Constants in Context

This lesson will apply the mathematical information learned in the previous lessons to real world situations. These problems demonstrate knowledge and interpretation of situations around us.

In order to model word problem situations, the words in the sentences must be understood.  In the examples, note that the letters used may vary. The variables x and y are used when graphing is involved so that they are easily related to a graph. Use what is appropriate to the problem.

Example 1:   A car has a gas tank that holds 18 gallons of gas. Juan has $50. How many gallons of gas will Juan be able to put in the gas tank if gas costs $2.50 a gallon? Does he have enough money to fill the gas tank? Will Juan have any money left over? If so, model the remaining money with a linear equality or inequality.

Step 1:  Associate the information given to you in the equation with variable letters. These do not have to be x and y but may be any letter that represents the information in the problem. Look for how many unknowns there are, mark with a ?

c = cost = $2.50 per gallon or 2.50/gallon

d = dollars available = $50

t  = gas tank size

g = gallons that may be bought?

Step 2: Write an equation associating the variables to find the missing one. Use the units, dollars and gallons to help determine the equation.

a. First, what unit is the final answer in? gallons

b. Write the equation.

gallons bought = dollars available/cost with units 

gallons bought = dollars available/cost{dollarspergallon}  

Step 3: Substitute values into the equation and solve for the missing value.  

g = 50 dollars/2.5 dollarsgallon  = 50 dollars/1  * 1 gallon/2.50 dollars  = 20 gallons

Step 4:  Answer the question.

Yes,  Juan could buy 20 gallons of gas which is more than enough to fill an 18 gallon gas tank.  Juan will have at least $5.00 left over if he has to fill the tank completely. Dollars remaining,  r ≥ $5.00.

Example 2: Fewer than 50 girls and boys came to the tryouts for the debate team.  Write an inequality in two variables to model the situation.

b = the number of boys

g = the number of girls

t = the total number of students < 50

b + g < 50    

Example 3:  To fix the refrigerator, a service company provides an estimate of at most $170.  The estimate includes a service charge of $75 to come out to the house and an hourly rate of $38. Write an inequality to model the situation. How many hours is estimated to repair the refrigerator?

c = service charge = $75

r = hourly rate = $38/hr

h = number of hours to repair the refrigerator?

t = the total cost of the repair ≤ $170

c + r * h ≤ t

$75 + $38h ≤ $170 Inequality Model

$38h ≤ $95

h ≤ 2.5 hours

 

Review

In this module we examined one and two variable linear equations, equalities and inequalities. Some special features of solving equations are extremely important.

Rules:  

image: Whatever you do on one side 

and when inequalities are involved

image: when multiplying or dividing by a negative number flip the inequality sign 

Additionally, there are three important formulas to find the equations of lines.

Slope formula              

m = y-y1/x-x1  

Slope-Intercept form

y = mx + b

Point-Slope form

y - y1 = m(x - x1)

 

Linear Equations were studied in algebraically and graphically and then used in contextual problems. This module is the fundamental building blocks for the remainder of the material for the SAT. It is imperative that a deep understanding of the relationship between algebraic and graphical representation and analysis exists.

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