Language of Algebra.

Presentazione sul tema: "Language of Algebra."— Transcript della presentazione:

1 Language of Algebra

2 Basic concepts Key words Practice exercises
Lesson 1 order Basic concepts Key words Practice exercises

3 Operation Addition (+) Subtraction (-) Moltiplication (.) Division (:)
Sum difference product quotient total less Double/triple Divide into plus minus times divide Added to subtract multiply Shared equally among More than Less than Multiply by per Increased by Decreased by of Divided by

4 Exercises together! A number increased by five X+5
Twice a number minus four 2x-4 Seven divided by a number 7 : x

5 Here are some examples that show how letters can be shorter than words
Here are some examples that show how letters can be shorter than words. Match the phrases with the letters that can be used in their place. A number plus two Three times a number Seven subtracted from a number A number divided by three A number taken away from seven Two different number added together A number multiplied by itself n-7 n/3 n+2 x+y a2 3n 7-n

6 Other examples! Five taken from a number x-5 A number less than two
Five less a number 5-x Four substracted from twice a number 2x-4 Now think of five more examples in words. Read our your examples and try to get your friend to write them as algebra. Be careful-It’s not easy!

7 We continue… The sum of 5 and a number The product of a number and 8
Twice the difference of a number and 7 Twice the number decreased by 6 5+x x or x 2(x-7) 2x-6

8 Set up an equation 3 times a number added to 4 is 16
The sum of -5 and a number is equal to 8 Twice a number less than 4 gives us 10 3x+4=16 -5+x=8 4-2x=10

9 Lessons 2 The linear equations

10 Finite number of solutions
Equation Finite number of solutions What’s it? determined Can be It ‘s an equality between two expressions containing one or more variables , called unknowns 2x+3=x+7 To solve an equation no solutions impossible indeterminate infinite ly many solutions has degree equal means finding the value of the unknown for which the equation becomes a true statement; a solution is said to satisfy the equation. to the degree of the polynomial p ( x ) I° degree 3x+1=2 II° degree 3 x2 +1=5 I° degree into two unknowns x+y=4

11 Two equations are equivalent if they are the same set of solutions.
First principle of equivalence: If you add or subtract the same quantity to both sides of an equation, you obtain an equation which is equivalent to the one you had in the beginning

12 Solving Linear Equations
METHOD #1 – Inspection Something added to 5 equals 12. You could start testing numbers until you get the answer or just “know” the answer will be seven. METHOD #2 – Isolation This method uses opposite operations. You will “isolate” the variable by getting it by itself on one side of the equation and the answer on the other side. OPPOSITE OPERATIONS : ADD SUBTRACT MULTIPLY DIVIDE

13 STEPS : 1. On which side of the equation is my unknown ?
2. If the unknown appears on both sides, you’ll need to get it on one side 3. Move all constants first 4. Move all coefficients last EXAMPLE 1 : Solve The left side = the right side. Acts like a seesaw. If I add 10 kg to one side, I have to add 10 kg to the other side to keep it balanced. Variable is on left side, so I will move things from the left side to the right side… STEPS : On which side of the equation is my unknown ? 2. If the unknown appears on both sides, you’ll need to get it on one side 3. Move all constants first 4. Move all coefficients last

14 +8 needs to move, so I will subtract 8 from both sides…
STEPS : On which side of the equation is my unknown ? 2. If the unknown appears on both sides, you’ll need to get it on one side 3. Move all constants first 4. Move all coefficients last You can see that now we have simplified the left side and our unknown is almost by itself. Remember, to be by itself, it needs to have a coefficient of one…

15 Our unknown is being multiplied by 2, so we will divide BOTH sides by 2…
STEPS : On which side of the equation is my unknown ? 2. If the unknown appears on both sides, you’ll need to get it on one side 3. Move all constants first 4. Move all coefficients last

16 Let’s check this answer back into the original equation…
STEPS : On which side of the equation is my unknown ? 2. If the unknown appears on both sides, you’ll need to get it on one side 3. Move all constants first 4. Move all coefficients last

17 Good Work!

18

19 Second principle of equivalence:
If you multiply or divide by the same quantity on both sides of an equation , you obtain an equation which is equivalent to that which it had in the beginning

20 EXAMPLE 2 : Solve Variable is on left side, so I will move things from the left side to the right side… Another way to attack the first step… Place a box around your variable, its coefficient, and any sign that is in front… What do you see outside the box on the same side of the equation ?…

21 ( - 4 )…so ADD 4 to both sides…
When the coefficient of your unknown is a fraction, multiply both sides by the reciprocal ( flip your fraction )…

22 EXAMPLE 3 : Solve Variable is on right side, so I will move things from the right side to the left side Place a box around your variable, its coefficient, and any sign that is in front… What do you see outside the box on the same side of the equation ?… ( + 9 )…so subtract 9 from both sides… Unknown is being multiplied by ( - 3 ) so we will divide both sides by ( - 3 )

23 EXAMPLE 4 : Solve OK, in this example the variable appears on both sides of the equation. We’ll need to get it on one side so we need to move one of them. I’m going to move the 7x from the right side to the left side… Move the 7x from the right side to the left side… ANSWER Place a box around your unknown and any coefficient and signs… You see +7, so subtract seven from both sides… The solution MUST be positive so divide both sides by ( - 1 )…

24 EXAMPLE 5 : Solve Get the variables on one side… Add 11 to both sides… Divide both sides by 5… You might get a fraction for an answer. Leave improper fractions as improper…

25 EXAMPLE 6 : Solve 1st - distribute Subtract 20 from both sides

26 Now solve you!

27 Now try it!

28 The consequences of these two principles :
You can transfer a term from one side of the equation to the other, just changing its sign: Two equal terms in both sides can be deleted:\ You can write an equation in the normal form by transferring all its terms at the first side, leaving a 0 in the second:

29 If the two sides of the equation have a common non-zero factor, this can be semplified:
If necessary, you can change the sign of all the terms of the equation in both sides: If the two sides are fractional expressions, you can remove the denominators by multiplying both sides by their L.C.D (which must be different from zero):

30 Good Job!

31 “Anything that does not condense into an equation is not science”
Albert Einstein

32 Videography: https://www.youtube.com/watch?v=bAerID24QJ0

33 Problems https://www.youtube.com/watch?v=xKH1Evwu150