1. Copyright 2013, 2010, 2007, 2005, Pearson, Education, Inc.

2. Formulas and Problem Solving
2.6
Formulas and Problem Solving

3. Formulas
A formula is an equation that states a known relationship among multiple quantities (has more than one variable in it)
A = lw (Area of a rectangle = length · width)
I = PRT (Simple Interest = Principal · Rate · Time)
P = a + b + c (Perimeter of a triangle = side a + side b + side c)
d = rt (distance = rate · time)
V = lwh (Volume of a rectangular solid = length · width · height)

4. Example
Charles Pecot can afford enough fencing to enclose a rectangular garden with a perimeter of 140 feet. If the width of the garden is to be 30 feet, find the length.
l = length of the garden
w = width of the garden = 30 feet
P= perimeter of the garden = 140 feet

5. continued
Charles Pecot can afford enough fencing to enclose a rectangular garden with a perimeter of 140 feet. If the width of the garden is to be 30 feet, find the length.

6. continued
Substitute 40 for l and 30 for w in the perimeter formula and check to see that the perimeter is 140 feet.
The length of the rectangular garden is 40 feet.

7. Example
A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side is 30 feet more than the length of the shortest side. Find the dimensions if the perimeter is 102 feet.
The formula for the perimeter of a triangle is P = a + b + c.
If we let
x = the length of the shortest side, then
2x = the length of the second side, and
x + 30 = the length of the third side

8. continued Formula: P = a + b + c Substitute: 102 = x + 2x + x + 30

9. continued
Check: If the shortest side of the triangle is 18 feet, then the second side is 2(18) = 36 feet, and the third side is
= 48 feet. This gives a perimeter of
P = = 102 feet, the correct perimeter.
State: The three sides of the triangle have a length of 18 feet, 36 feet, and 48 feet.

10. Solving a Formula for a Variable
It is often necessary to rewrite a formula so that it is solved for one of the variables.
To solve a formula or an equation for a specified variable, we use the same steps as for solving a linear equation except that we treat the specified variable as the only variable in the equation.

11. Solving Equations for a Specified Variable
Step 1: Multiply on both sides to clear the equation of fractions if they appear.
Step 2: Use the distributive property to remove parentheses if they appear.
Step 3: Simplify each side of the equation by combining like terms.
Step 4: Get all terms containing the specified variable on one side and all other terms on the other side by using the addition property of equality.
Step 5: Get the specified variable alone by using the multiplication property of equality.

12. Example
Solve T = mnr for n.

13. Example
Solve for A = PRT for T.

14. Example Solve for P. Factor out P on the right side.
Divide both sides by 1 + RT.
Simplify.