1.3 Modeling with Linear Equations

A. Translate key words and phrasesB. Application problems involving percentages

C. Application problems using common formulasD. Solve a formula for a specific variable

A. “The sale price is $10 less than the list price L”

• WRONG WAY: S = 10 – L • RIGHT WAY: S = L – 10

• Try: “Three less than z”

• Review other key words/phrases on p. 99

B. Word problems with %s

• You take a job where you’ll get paid $8 an hour. You are promised a raise in 2 months, after which you will be paid $9 an hour. What percentage raise is this?

• STRATEGY: reword the question:

• “$1 is what percent of $8?”

70 is 40% of what number?

70is 40% what numberof

C. Formulas you’ll need

• If we expect you to know any of these on exams, it will most likely be those commonly known, like

• perimeter,

• area,

• distance = rate times time

• those related to cost, revenue, and profit

• All formulas are listed on p. 103

• “Write an algebraic expression for the distance traveled in t hours by a car traveling at 40 miles per hour.”

• FORMULA TO KNOW: distance = rate times time

• The formula gives us a structure ___=(___)(___) into which we can put what information we have been given to form the equation we need. 40 miles per hour is a rate. I can tell this by the units “miles per hour.” Time will be indicated by t. I can tell because it says “t hours.”

distance = (rate)(time)

distance = (40)(t)

Notice that the instructions say to write an algebraic EXPRESSION, not an EQUATION, so I should not make “distance = 40t” my answer.

ANSWER: 40t

• Write an expression for the total revenue from selling x units at $3.59 per unit.

• Revenue = (price per unit) x (# of units)

• Revenue = (3.59)(x)

• But they want an expression, not an equation:

• Answer: 3.59x

I was a REALTOR, and used this kind of algebra all the time:

• You are buying a home. The real estate agent wants 3% and the seller wants $100,000. How much must you write a check for to meet everyone’s demands? (CAUTION: The agent wants 3% of the check amount, not 3% of $100,000.)

• If an agent jumps to conclusions and says $103,000, that will NOT be enough, the seller will get angry at the closing table and it will be a disaster. Moral: Get a realtor who can do algebra for sure. See next slide…

Seller wants 100K, agent wants 3%...• This one may not come out evenly, but we’ll round to the nearest penny in the end.• There isn’t a formula in the book, but there is a formula that we can start with that

comes from our knowledge of how commission works:

PRICE – COMMISSION = WHAT SELLER GETS

Since we know that commission in this case is supposed to be 3% of the price, we can actually write this as:

PRICE – (3% of PRICE) = WHAT SELLER GETS

Moreover, in this problem, the seller clearly wants to get 100K, so:

PRICE – (3% of PRICE) = 100,000

The price is the unknown that we ultimately want to find. Let’s call it x:

x – (3% of x) = 100,000

For percents, move decimal two places, and “of” means times:

1x – (0.03x) = 100,000

One x minus 0.03 x will be 0.97 x.

0.97x = 100,000

x = 100,000 divided by 0.97

x = 103,092.78

Luckily, your agent can do algebra and will ensure that the seller is happy with 100K without making you pay a penny more than necessary. $103,092.78 would be the price.

D. Solving for a variable

Solve for f: H = (1/3)(g + f)k,[suppose k is not zero]

As I said the other day in class, when solving, you will go in the opposite order you would go if you were doing computation. For example, if you were plugging in a value for f, like plugging in the number one for f, you would first add it to g, and then you would multiply the result by 1/3 and by k. But in this case, since we are solving, we will UNDO those things in the opposite order. The first shall be last, and the last shall be first. FIRST, I’ll work on undoing the 1/3 or the k since those two pieces were the last things you would do computationally. To me, it feels like I am shaving off the outside layors, one by one, until the only thing that will remain will be the f.

Solve for A: B = A + NA• The idea of “getting A by itself” is a

problem because there are two A’s in this one. So we need to get the A’s together AND get A by itself.

• Many of you would probably think of the factor-it-out strategy, but implementing that correctly is tricky. Think, or even show, outside and under, to make sure you factor the A out correctly.

Solve for A: B = A + NA

Solve for w: I = B + Bwr• Because students get so excited about the

factor-it-out strategy, many are surprisingly tempted to factor out the B because the structure of this equation looks so similar to the last one we did; however, this one is not asking us to solve for B. Therefore, the fact that there are two B’s here instead of one B does not bother me at all. The instructions say to solve for w. I like to draw an arrow on the thing I’m solving for, so I don’t get confused or forget.

Solve for w: I = B + Bwr

Solve for b: A = d + (b – 1)n

Many people have this overwhelming desire to distribute the n first, but that is not necessary here. You could do it, but it is not a must; it’s kind of an extra step. I’ll show you how I would solve for b without distributing the n.

Computationally, the order of things done to b was subtract one, multiply by n, and then add d.

Therefore, the order I “undo” all that will be reversed: subtract d, divide by n, and then add 1.

Solve for b: A = d + (b – 1)n