2-8 Literal Equations and Dimensional Analysis A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 2-8 Literal Equations and Dimensional Analysis Objectives: Solve equations for a given variable. Use formulas to solve real-world problems

Each year more people use credit cards to make everyday purchases Each year more people use credit cards to make everyday purchases. If the entire balance is not paid by the due date, compound interest is applied. The formula for computing the balance of an account with compound interest added annually is 𝐴=𝑃 (1+𝑟) 𝑡 𝐴= the amount of money in the account including the interest. 𝑃= the amount in the account before interest is added 𝑟= the interest rate written as a decimal 𝑡= the time in years Suppose you know the quantities for A and r, but not P. How would you solve for P? Some equations like the one above contain more than one variable. At times, we will need to solve for one of the variables.

Example 1: Solve for the specific variable.

Example 1: Continued… c) 28=𝑡 𝑟+4 , 𝑓𝑜𝑟 𝑡 d) 𝑎 𝑞−8 =23, 𝑓𝑜𝑟 𝑞

You try… e) 5𝑏+12𝑐=9, 𝑓𝑜𝑟 𝑏

In some cases we start with the variable we are trying to solve for on both sides. In this case we want the variable we are solving for all on one side and in some cases we may use the distributive property

Example 2: Solve each equation for the variable indicated.

You try… c) 7𝑥−2𝑧=4−𝑥𝑦, 𝑓𝑜𝑟 𝑥

Literal Equation An equation that involves several variables. Also called a formula

TRIVIA… 1. Which NFL team lost four Super Bowls in a row? Buffalo Bills 1.  Which NFL team lost four Super Bowls in a row? 2.  What state does Peter Griffin from "Family Guy" live in? Rhode Island 3.  Who played The Joker in the movie "The Dark Knight?" Heath Ledger

Example 3: The formula for the volume of a rectangular prism is 𝑣=𝑙𝑤ℎ, where 𝑙= length, 𝑤= width, & ℎ= height. a) Solve the formula for 𝑤. b) Find the width of a rectangular prism that has a volume of 79.04 𝑐𝑢𝑏𝑖𝑐 𝑐𝑚, a length of 5.2 𝑐𝑚, & a height of 4 𝑐𝑚.

You try… A car's fuel economy E (miles per gallon) is given by the formula 𝐸= 𝑚 𝑔 , where 𝑚 is the number of miles driven and 𝑔 is the number of gallons of fuel used. a) Solve the formula for 𝑚. b) If Sophia’s car has an average fuel consumption of 30 𝑚𝑖/𝑔𝑎𝑙 and she used 9.5 𝑔𝑎𝑙. How far did she drive?

Dimensional Analysis or Unit Analysis Process of carrying units throughout a computation We can convert one unit into another by multiplying by a ratio. Example: 47 𝑚𝑖 ℎ𝑟 𝑖𝑛𝑡𝑜 𝑓𝑡 𝑠𝑒𝑐 𝑜𝑟 𝑖𝑛𝑡𝑜 𝑦𝑑 𝑚𝑖𝑛 We do this so our unit rates will make more sense in the situation.

Conversions to memorize 12 in = 1 ft 3 ft = 1 yd 5280 ft = 1 mi 60 sec = 1 min 60 min = 1 hour ∎ ∆ ∙ ∆ 𝛼 = 3 7 ∙ 7 4 =

Steps for Unit Analysis Write as a fraction. Look at what units are changing and what is staying the same. Find the conversion and write it as a ratio so the units will cancel. Multiply across the top and across the bottom, and then divide.

Example 4: Convert. a) 35 𝑚𝑖 ℎ𝑟 → 𝑚𝑖 𝑚𝑖𝑛 b). 82 𝑓𝑡 𝑚𝑖𝑛 → 𝑖𝑛 𝑚𝑖𝑛

Example 4: continued… c) 135 𝑖𝑛 𝑚𝑖𝑛 → 𝑖𝑛 ℎ𝑟 d)193 𝑓𝑡 𝑚𝑖𝑛 → 𝑚𝑖 ℎ𝑟

Practice: WS 2-8

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