2-5 Equations and Problem Solving; 2-6 Formulas

Defining One Variable in Terms of Another  The length of a rectangle is 6 in. more than its width. The perimeter of the rectangle is 24 in. What is the length of the rectangle?  RELATE: The length is 6 in. more than the width.  DEFINE: Let w=the width; w+6=the length  The length is described in terms of the width. So define a variable for the width first.

Defining One Variable in Terms of Another

 The width of a rectangle is 2 cm less than its length. The perimeter of the rectangle is 16 cm. What is the length of the rectangle?

Consecutive Integers  Consecutive Integers differ by 1. The integers 50 and 51 are consecutive integers, and so are -10, -9, and -8. For consecutive integer problems, it may help to define a variable before describing the problem in words. Let a variable represent one of the unknown integers. Then define the other unknown integers in terms of the first one.

 The sum of three consecutive integers is 147. Find the integers.  DEFINE:  Let n=the first integer  Then n+1=the second integer  And n+2=the third integer  RELATE:  First Integer + Second Integer + Third Integer=147  WRITE:  n+(n+1)+(n+2)=147 Consecutive Integers

 The sum of three consecutive integers is 48.  Define a variable for one of the integers.  Write expressions for the other two integers.  Write and solve an equation to find the three integers.

Same Direction Travel  A train leaves a train station at 1 P.M. It travels at an average rate of 60 mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 96 mi/h. The second train follows the same route as the first train on a track parallel to the first. In how many hours will the second train catch up with the first one?  DEFINE:  Let t=the time the first train travels  Then t-1=the time the second train travels  RELATE

Same Direction Travel

Round Trip Travel  Nina drives into the city to buy a software program at the computer store. Because of traffic conditions, she averages only 15 mi/h. On her drive home she averages 35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the computer store?  DEFINE:  Let t=time of Nina’s drive to the computer store  2-t=the time of Nina’s drive home.  RELATE:  Nina drives 15t miles to the computer store and 35(2-t) miles back

Round Trip Travel

 On his way to work from home, your uncle averaged only 20 miles per hour. On his drive home, he averaged 40 miles per hour. If the total travel time was 1 ½ hours, how long did it take him to drive to work?

Opposite-Direction Travel  Jane and Peter leave their home traveling in opposite directions on a straight road. Peter drives 15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find Peter’s rate and Jane’s rate.  DEFINE:  Let r=Jane’s rate  Then r+15=Peter’s rate  RELATE:  Jane’s distance is 3r. Peter’s distance is 3(r+15)  Remember that the sum of their distances is the total distance! WRITE: 3r+3(r+15)=225

Opposite-Direction Travel

 Sarah and John leave Perryville traveling in opposite directions on a straight road. Sarah drives 12 miles per hour faster than John. After 2 hours, they are 176 miles apart. Find Sarah’s speed and John’s speed.

2-6 Formulas  A literal equation is an equation involving two or more variables. Formulas are special types of literal equations. To transform a literal equation, you solve for one variable in terms of the others. This means that you get the variable you are solving for alone on one side of the equation.

Transforming Geometric Formulas

Transforming Equations  Solve y=5x+7 for x.  Subtract 7 from each side  Simplify  Divide each side by 5  Simplify

Transforming Equations Containing only Variables  Sometimes an equation will only have variables. Transforming this type of equation is no different from transforming equations with numbers.  Solve ab-d=c for b.  Add d to each side  Combine like terms  Divide each side by a, a≠0  Simplify