Let’s start with a riddle. Find a number such that when you add 5 to it, the number triples.
To solve this riddle, let’s call the unknown number x. Adding 5 to it produces x+5. Tripling the original number gives us 3x. We want those numbers to be equal, so we have to solve the equation
3x=x+5
If we subtract x from both sides of the equation, we get 2x=5
(Where does 2x come from? 3x−x is the same as 3x−1x, which equals 2x.) Dividing both sides of that equation by 2 gives us
x=5/2=2.5
We can verify that this answer works, since 2.5+5 = 7.5, which is the same as 3 times 2.5.
Aside
Here’s another trick that algebra can help us explain. Write down any three-digit number where the digits are in decreasing order, like 842 or 951. Then reverse those numbers and subtract the second number from the first. Whatever your answer is, reverse it, then add those two num-bers together. Let’s illustrate with the number 853.
853 495
− 358 +594
495 1089
Now try it with a different number. What did you get? Remarkably, as long as you follow the instructions properly, you will always end up with 1089! Why is that?
Algebra to the rescue! Suppose we start with the three-digit number abc where a>b>c. Just as the number 853= (8×100) + (5×10) +3, the number abc has value 100a+10b+c. When we reverse the digits, we get cba, which has value 100c+10b+a. Subtracting, we get
(100a+10b+c)−(100c+10b+a)
= (100a− a) + (10b− 10b) + (c− 100c)
=99a− 99c=99(a− c)
In other words, the difference has to be a multiple of 99. Since the origi-nal number has digits in decreasing order, a−c is at least 2, so it must be 2, 3, 4, 5, 6, 7, 8, or 9. Consequently, after subtracting, we are guaranteed to have one of these numbers:
198, 297, 396, 495, 594, 693, 792, or 891
In each of these situations, when we add the number to its reversal, 198+891=297+792=396+693=495+594=1089 we see that we are forced to end up with 1089.
We have just illustrated what I call the golden rule of algebra: do unto one side as you would do unto the other.
For example, suppose you wish to solve for x in the equation 3(2x+10) =90
Our goal is to isolate x. Let’s begin by dividing both sides by 3, so the
equation simplifies to
2x+10=30
Next, let’s get rid of that 10 by subtracting 10 from both sides. When we do that, we get
2x=20
Finally, when we divide both sides by 2, we are simply left with x=10
It’s always a good idea to check your answer. Here, we see that when x = 10, 3(2x+10) = 3(30) = 90, as desired. Are there any other solutions to the original equation? No, because such a value of x would also have to satisfy the subsequent equations, so x = 10 is the only solution.
Here’s a real-life algebra problem that comes from the New York Times, which reported in 2014 that the movie The Interview, produced by Sony Pictures, generated $15 million in online sales and rentals dur-ing its first four days of availability. Sony did not say how much of this total came from $15 online sales versus $6 online rentals, but the studio did say that there were about 2 million transactions overall. To solve the reporter’s problem, let’s let S denote the number of online sales and let R denote the number of online rentals. Since there were 2 million transactions, we know that
S+R=2,000,000
and since each online sale is worth $15 and each online rental is worth
$6, then the total revenue satisfies
15S+6R =15,000,000
From the first equation, we see that R = 2,000,000−S. This allows us to rewrite the second equation as
15S+6(2,000,000− S) =15,000,000
or equivalently, 15S+12,000,000−6S = 15,000,000, which only uses the variable S. This can be rewritten as
9S+12,000,000=15,000,000
Subtracting 12,000,000 from both sides gives us 9S =3,000,000
and therefore S is approximately one-third of a million. That is S ≈ 333,333 and so R = 2,000,000−S ≈ 1,666,667. (Checking: total sales would be $15(333,333) +$6(1,666,667) ≈$15,000,000.)
It’s time to discuss a rule that we have been using throughout this book without explicitly naming it, called the distributive law, which is the rule that allows multiplication and addition to work well together.
The distributive law says that for any numbers a, b, c, a(b+c) =ab+ac
This is the rule that we are using when we multiply a one-digit number by a two-digit number. For example,
7×28 =7×(20+8) = (7×20) + (7×8) =140+56=196 This makes sense if we think about counting. Suppose I have 7 bags of coins, and each bag has 20 gold coins and 8 silver coins. How many coins are there altogether? On the one hand, each bag has 28 coins, so the total number of coins is 7×28. On the other hand, we can also see that there are 7×20 gold coins and 7×8 silver coins, and therefore (7×20) + (7×8)coins altogether. Consequently, 7×28 = (7×20) + (7×8).
You can also view the distributive law geometrically by looking at the area of a rectangle from two different perspectives, as in the picture below.
a
b c
ab ac
The rectangle illustrates the distributive law:a(b+c) =ab+ac
On the one hand, the area is a(b+c). But the left part of the rectan-gle has area ab and the right part has area ac, so the combined area is
ab+ac. This illustrates the distributive law whenever a, b, c are positive numbers.
By the way, we sometimes apply the distributive law to numbers and variables together. For instance,
3(2x+7) =6x+21
When this equation is read from left to right, it can be interpreted as a way of multiplying 3 times 2x+7. When the equation is read from right to left, it can be seen as a way of factoring 6x+21 by “pulling out a 3”
from 6x and 21.
Aside
Why does a negative number times a negative number equal a positive number? For instance, why should(−5) × (−7) = 35? Teachers come up with many ways to explain this, from talking about canceling debts to simply saying “that’s just the way it is.” But the real reason is that we want the distributive law to work for all numbers, not just for positive numbers. And if you want the distributive law to work for negative numbers (and zero), then you must accept the consequences. Let’s see why.
Suppose you accept the fact that −5×0 = 0 and−5×7 = −35.
(These can be proved too, using a strategy similar to what we are about to do, but most people are happy to accept these statements as true.) Now evaluate the expression
−5 ×(−7+7)
What does this equal? On the one hand, this is just−5×0, which we know to be 0. On the other hand, using the distributive law, it must also be((−5) × (−7)) + (−5×7). Consequently,
((−5)×(−7)) + (−5 × 7) = ((−5)×(−7))− 35=0
And since ((−5) × (−7)) −35 = 0, we are forced to conclude that (−5) × (−7) = 35. In general, the distributive law ensures that (−a) × (−b) =ab for all numbers a and b.