DEMO LESSON
An equation is a sentence that states that two algebraic expressions are equal. These expressions are called left side and right side of the equation. |
Equation may be a true sentence such as 2 + 3 = 5 , a false sentence such as 2 + 3 = 7 , or an open sentence like 3x + 2 = 7 . Open sentence contains a variable. We cannot say whether it is true or false, because a variable can have different values.
To solve an equation means to find a value of the variable that makes this equation a true sentence. This value is called a solution, or a root of the equation.
Equations may have one root, more than one root, or no roots at all. For now we will deal only with equations that have exactly one root.
Equation is a wonderful kind of problem: you can always check yourself whether you got it right. To do it, simply replace the variable in the initial equation with your solution and find the values of the right and left sides of the equation. If they are equal, you are correct. Make it a habit to check your solutions.
To solve an equation we try to simplify, step by step, both right and left sides of the equation using properties of operations. Each new simplified equation is equivalent to the previous one. That means it has the same root. We write all the equations down one under another. The process is finished with the equation x= c, where c is the root.
So far to solve equations we needed just one step, as in these examples:
1) x = b a | ax = b | 2) x = ab | a x = b | 3) | x a = b |
x = a b | |||||
4) x = b - a | x + a = b | 5) x = a - b | x - a = b | 6) | a - x = b |
x = a + b |
Now let us try to solve harder equations:
Equation | What you can do |
---|---|
Examlpe 1. | |
3x + 2x = 10 | Collect like terms |
5x = 10 | |
x = 2 | Check: 3 2 + 2 2 = 6 + 4 = 10 - true sentence. |
Examlpe 2. | |
7x + 6 = 111 | You know how to solve equations x+a=b. Think of 7x as your variable. |
7x = 111 - 6 | Find what 7x equals to. |
7x = 105 | |
x = 1057 | |
x = 15 | Check: 7 15 + 6 = 105 + 6 = 111 - true sentence. |
Examlpe 3. | |
(2x - 7)5 = 65 | You know how to solve equations a x = b. First find what 2x + 7 equals to. |
2x - 7 = 65 5 | |
2x -7=325 | You know how to solve equations x- a=b. First find what 2x equals to. |
2x = 325 + 7 | |
2x = 332 | |
x = 332 2 | |
x = 166 | Check: (2 166 - 7) 5 = (332 - 7) 5 = 325 5 = 65 - true sentence. |
Examlpe 4. | |
3(d + 2) - 2d = 9 | First remove the brackets according to the distributive law of multiplication. |
3d + 6 - 2d = 9 | Collect like terms. |
d + 6=9 | |
d = 9 - 6 | |
d = 3 | Check: 3 (3 + 2) - 2 3 = 15 - 6 = 9 - true sentence. |
Exercises.
1.1. Determine whether the given value is a root of the equation. (Just put the number instead of the variable in the equation, and see if the equation is true).
a) |
21x + 53 = 190; |
x = 7 |
b) |
25y - 3 = 97; |
y = 4 |
c) |
3(d + 1) + 3d = 3; |
d = 7 |
1.2. Solve the equations:
a) |
(x + 7) 7 = 11 |
b) |
16(2y - 8) = 128 |
c) |
25 + 17x - 16x = 74 |
d) |
3x + x + 2x - 56 = 72 |
1.3. Try to write an equation based on the information given, then solve it:
- The sum of 3 and some number equals twice as much as 11. Find the number.
- The difference of some number and 19 is 6. Find the number.
- Three times a number plus the same number equals 7. Find the number.
1.4. Solve the equations:
a) |
x + x + x = 255 |
b) |
13z - 2z + 3z = 336 |
c) |
x + 163 + 17 = 256 |
d) |
15y - 6y + 4y = 221 |
1.5. Solve the equations:
a) |
4x + x - 19 = 31 |
b) |
6x + 17 + 4x + 13 = 106 |
c) |
12x - 16 + 8x = 84 |
d) |
2x + 4x + 21 - 13 = 80 |
1.6. Solve the equations:
a) |
4(x + 5) = 48 |
b) |
3(n + 7) = 45 |
c) |
(x - 2) 3 = 15 |
d) |
4(n + 3) = 236 |
1.7. Solve the equations:
a) |
2(x + 7) + 5 = 111 |
b) |
3(y - 8) + 7 = 78 |
c) |
4(b + 3) - b = 12 |
d) |
2(x + 3) + 5(x + 3) = 364 |
1.8. Try to write an equation based on the information given, then solve it:
- One half of a number, decreased by 8, equals 12. Find the number.
- Twice a number, increased by 13, equals 27. Find the number.
- The sum of four times a number and 7 equals 43. Find the number.
Puzzle of the week
1.9. Different letters stnd for different digits. Sole he puzzle.
(E + Q) ÷ U = A - T = I ÷ O = N
II. Fibonacci numbers 1, 1, 2, 3, 5, 8, 13...
The Fibonacci numbers are named ater Leonardo of Pisa, known as Leonardo Fibonacci (Fi-bonacci means "son of Bonacci"), although the numbers had been described earlier in India. Leonardo Fibonacci (1170 - 1250) was an Italian mathematician. He played an important role in reviving ancient mathematical skills of East in Europe and made his own contributions to mathematics. Leonardo's father was a diplomat. He travelled a lot and took his son with him to Mideterranian countries. As a child, Leonardo studied math in North Arfrica. He learned a lot of things that were not in use in Europe. After he returned to Italy, he wrote books on Arithmetic, Algebra and Geometry. "Liber abaci" , his most famous book, introduced the Hindu-Arabic number system into Europe. Before Fibonacci, everyone used Roman numerals in Italy! |
Fibonacci is best remembered today for the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... in which each number is the sum of the two preceding numbers. It appears in many different areas of mathematics and science.Fibonacci numbers appear in nature as well. For example,
1) Petals on flowers:
On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21, whereas daisies can be found with 34, 55 or even 89 petals. |
2) Pine cones, pineapples, cacti, cauliflower and broccoli.
The numbers of rows (spirals) that go in one direction - are Fibonacci numbers. The numbers of rows in the three directions on the pineapple are 8, 13, and 21. The numbers of rows in 2 different directions on pine cones - 8 and 13. |
3) Leaf arrangement:
Some common trees with their Fibonacci leaf arrangement numbers are: 1/2 elm, lime, grasses; 1/3 beech, hazel, grasses, blackberry; 2/5 oak, cherry, apple, holly, plum; 3/8 poplar, rose, pear, willow; 5/13 almond, |
where t/n means each leaf is t/n of a turn after the last leaf or that there is there are t turns for n leaves.
The Fibonacci numbers form a sequence defined by the following recurrence relation : F_{0} = 0; F_{1} = 1; F_{n} = F_{n-1} + F_{n-2} for n > 1.
That is, after two starting values 0 and 1, each Fibonacci number is the sum of the two preceding Fibonacci numbers.
The first Fibonacci numbers, also denoted as F_{n}, for n = 0, 1, 2,… , are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418…
Sometimes Fibonacci sequence is considered to start at F_{1} = 1, but it is more common to include F_{0} = 0.
Example. Somebody bought a pair of rabbits and placed them into a cage. Every month this pair gives a birth to two new rabbits. A pair of new rabbits starts to produce rabbits when they are two months old. How many rabbits will there be in a year? In two years?
Solution.
1. Let R_{k} be the number of rabbits after month k.
2. Then R_{0} = 2, R_{1} = 4, R_{2} = 6. After three months we shall have R_{3} = 10 (pair of new rabbits that were born in the first month produced two new rabbits), and so on.
3. We shall prove by induction that R_{k} = 2F_{k+2}.
4. The induction statement for k = 1, 2, 3 was already tested: R_{1} = 4 = 2 F_{3} , R_{2} = 6 = 2 F_{4} , R_{3} = 10 = 2 F_{5} .
5. If after the month 1, 2 , …, k the number of rabbits is R_{1}, R_{2},… R_{k} = 2F_{k+ 2 then }after month k the number of productive rabbits is R_{k-2} = 2F_{k}.
6. Then after month (k + 1) the number of rabbits will be R_{k+1} which is: R_{k} (number of rabbits which already were after the month k) plus R_{k-1} (children of productive rabbits), or R_{k+1} = R_{k} + R_{k-1}= 2F_{k+2} + 2F_{k+1} = 2F_{k+3}.
7. We have: R_{12} = 2F_{14} = 2*377 = 754, R_{24} = 2F_{26} = 2*121393 = 242786.
Exercises.
1.10. Craig is going up the staircase consisting of 10 steps. His every move is either on the next step or on the step after next. How many ways are there to climb the staircase in such a manner?
1.11. How many ways are there to break the bill of $20 by the bills of $1 and $2 ?
1.12. How many ways are there to break the bill of $100 by the bills of $10 and $5?