Algebra Notes Writing Algebraic Expressions Let Statement math

Algebra Notes

Writing Algebraic Expressions

¬Let Statement: math sentence used to define a variable to represent the unknown quantities.

1. Laura has twice as much homework as Let Ann = a Ann. Let Laura = 2 a games lostmore =g 2. The Bills. Letwon five games than Let Bills won = 5 + g they lost. Let Yankees = y Let Tigers = 3 y 3. Seven more than three times a number is 25. Let width = w Let length = 3 + w 4. The length of a rectangle is 3 cm more

5. Mike is three years older than Jim. Let Jim = j Let Mike = 3+ j 6. Eight more than twice a number is 32 Let number = n 2 n + 8 = 32 7. Seven more than three times a number Let number = n is 25. 3 n + 7 Let numberincreased =n 8. Twice a number by four is 2 n + 4 = 16 16.

9. Six less than three times a number is Let number = n 21. 3 n – 6 = 21 Let than numbertwice =n 10. Fifteen less a number is 25. 2 n – 15 = 25 11. Sixty-six is eleven more Let number = n than five times a number. 66 = 5 n + 66

Writing Algebraic Expressions

Setting up & Solving word problems

¬Write your let statement ¬Write your equation ¬Solve ¬Check ¬Write an answer sentence

1. A cell phone company charges $39 a month plus $. 15 per text message sent. If Jan sends 35 text messages this month, how much does she owe before taxes are added? Let text message = t 39 + 0. 15 t t = 44. 25 Jam owes $44. 25 2. The Bills won five more= sgames than Let text message they lost. 12 + 2 s 4 snacks s=4

3. A rental car company ABC charges $25 per day plus $. 15 per mile. Rental car company XYZ charges $18 per day plus $. 25 per mile. If you plan to drive 50 miles, Let miles = m who is the cheaper rental company? XYZ is cheaper 4. ABC: 25 + 0. 15 m XYZ: 18 + 0. 25 $32. 50 $30. 50 Joe attends a carnival. The admission is $48. Tickets for rides cost $4 each. Joe needs one ticket for each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can number or rides = r the admission buy if he has. Let $200 before he pays fee. 48 + 4 r = 200 38 rides r = 38

substitutions

Evaluate if s = 4 1. 4 s → 4(4) 16 2. 4 + s → 4+4 8 3. 5 – s → 5 -4 1 4. 12 ÷ s → 12 ÷ 4 3

Evaluate if s = -6 1. 7 s → 7(-6) -42 2. 3 + s → 3 + (-6) -3 3. 7 – s → 7 – (-6) 13 4. 18 ÷ s → 18 ÷ (-6) -3

Evaluate if n = 3 and r = 5 1. n² + 7 r 3² + 7(5) 9 +12 21 2. 9 n - r² 9(3) - 5² 27 – 25 2 3. 2 nr + 6 n 2(3)(5) + 6(3) 30 + 18 48

Evaluate if p = 12 and q = -8 1. p + q +6 12 + (-8) + 6 -4 + 6 2 2. p – q + 3 12 – (-8) + 3 20 + 3 23 3. p – q + q² 12 – (-8) + (-8)² 20 + 64 84

Evaluate if a = -2 and b = 6 1. 3 a² + 5 b² 3(-2)² + 5(6)² 3(4) + 5(36) 192 2. 4 a³ + 3 b 4(-2)² + 3(6) 4(-8) + 18 -14 3. 7 a² - (b²/3) 7(-2)² - (6²/3) 7(4) – (36/3) 28 – 12

Like Terms

Terms of an Expression ¬Terms are parts of a math expression separated by addition or subtraction signs. 3 x + 5 y – 8 has 3 terms.

Like Terms ¬Like Terms: have the same variables to the same powers 8 x²+2 x²+5 a +a 8 x²and 2 x² are like terms 5 a and a are like terms

LIKE terms: Yes or No? 3 x + 7 x Yes - Like 5 x + 5 y No - Unlike 4 c + c Yes - Like 4 d + 4 No - Unlike

LIKE terms: Yes or No? 3 ab – 6 b No - Unlike 2 a – 5 a Yes - Like x and x² 6 and 10 No - Unlike Yes - Like

Identify the LIKE terms 3 m – 2 m + 8 – 3 m + 6 5 x + b – 3 x + 4 + 2 x – 1 – 3 b -6 y + 4 yz + 6 x² + 2 yz – 4 y + 2 x² - 5

Coefficients ¬A Coefficient: a number written in front of the variable. Example: 6 x The coefficient is 6. Example: x The coefficient is 1.

Simplify ¬Simplify: means to combine like terms. • Combine LIKE terms by adding their coefficients.

Write an expression: + 3 c + 4 c =7 c

Write an expression: 8 a - 1 a = 7 a

Write an expression: + 5 c + 4 d

Write an expression: 5 a – 4 b This expression cannot be simplified. Why not?

Simplify the following expressions

1. 2 x + 4 x 6 x 4. 2 a + 5 a + 7 a + 6 6 2. 3 a + 7 a 10 a 5. 3½y + 5 y 4½y 4 y 3. 6 xy – 2 xy 4 xy 6. 5 d – 6 d – 3 d -4 d 9. 4 s – 4 s 2 xy + 2 x 7. 3 xy – xy +2 x -2 c 10. -4 c + 8 c – 5 cd – 2 a 8. cd + 4 cd – 2 a -¾e 11. ½e – 2 e + ¾e 0 12. 5 x + 4 x + 11 x 24 x

Challenge questions

11. 3 x + 3 y + x + y + z 12. 5 b + 6 b² - 10 – 3 b 4 x + 4 y +z 6 a² + 7 b 10 13. Find the perimeter of the rectangle: A 4 x + 3 y B 8 x + 6 y C 12 xy D 4 x²+ 3 y²

Adding & subtracting Polynomials

Adding ¬Combine like term ¬Add the coefficients to simplify Example: ¬Start with: Add 2 x² + 6 x + 5 and 3 x² - 2 x – 1 2 x² + 6 x + 5 2 x² 3 x² ¬Place like terms together: 5 x² ____ + 3 x² - 2 x – 1 6 x – 2 x 5– 1 _______+ 4 x 4 5 x²_____+ + 4 x + 4 ¬Add the like terms: _____+ _____

Subtracting ¬Change the subtraction sign to addition and reverse the sign of each term that follows ¬Then add as usual Example: Subtract 5 y² + 2 xy - 5 and + + 3 x² - 2 x – 1 ¬Start with: 5 y² + 2 xy 5 y²- 53 y² 2 y² – -5 + 3 -2 xy 2 y² - 3 xy +3 3 xy -2 ¬Place like terms together: _______+ ________ 3 y² - xy - 2 ¬Add the like terms: _________+ _____

Try the following

1. (2 x + 3 y) + (4 x + 9 y) 6 x + 12 y 2. (3 a + 5 b + 7 c) - (5 a – 2 b + 9 c) -2 a + 7 b – 2 c 3. (3 x – 5) + (x – 11 x 7) + (7 x + 12) 4. (3 a + 5 b + 11 a 7 c) ++3 b (8 a – 2 c– 2 b – 9 c) 5. – 4 x³ + 6 x² – 8 x – 3 x³ 10 +and 2 x² 7 x³ - x -– 4 x² + 9 x + 3 7 6. Subtract (5 m – 6 n + 12) from (2 m + 3 n – 5) - (5 m – 6 n + 12) -3 m + 9 n -17

7. Subtract 8 a + 5 b – 6 c from 10 a + 8 b + 7 c (10 a + 8 b + 7 c) - (8 a + 5 b – 6 c) 2 a – 3 b + 13 c 8. (4 x + 8 y + 9 z – 7 a + 5 b) – (4 b + 5 x + 7 y + 3 z + 2 a) -x – y + 6 z – 9 a + b 9. (– 3 x 2 + 4 x – 11) – (– 6 x 2 – 8 x + 10). 3 x² + 12 x - 21 10. (7 e² + 3 e +2) + (9 – 6 e + 4 e²) + (9 e + 2 – 6 e²) 5 e² + 6 e + 13

challenge

Some of the measures of the polygons are given. P represents the measure of the perimeter. Find the measure of the other side or sides. x² - 15 x + 3 2 x + y 4 x - 3 14 x² - 4 x + 7

The distributive property

The Distributive Property ¬Distributive Property: the process of distributing the number on the outside of the parentheses to each term in the inside. a(b + c) = ab + ac Example: 5(x + 7) = 5 • x + 5 • 7 5 x + 35

Practice #1 3(m - 4) 3 • m-3 • 4 3 m – 12 Practice #2 -2(y + 3) -2 • y + (-2) • 3 -2 y + (-6) -2 y - 6

Simplify the following: 3(x + 6) = 3 x + 18 4(4 – y) = 16 – 4 y 7(2 + z) = 14 + 7 z 5(2 a + 3) = 10 a + 15

Simplify the following: 6(3 y - 5) = 18 y – 30 3 +4(x + 6) = 4 x + 27 2 x + 3(5 x - 3) + 5 = 17 x – 4

Distributive practice

+ c) 8 + 18 x 4. 7(a + c + b) 7 a + 7 c + 7 b 7. -1(x + 2) -x – 2 10. y(1 x) y ++yx 7 x - 7 12 a + 12 b + 2. 12 c 5. -10(3 + 2 + 7 x) 6. -1(3 w + 3 x + -2 z) -3 w – 3 x + -70 x - 50 2 z 8. 3(-2 + 2 x 2 y 3 + 3 y 2) 9. 5(5 5 x) + 25 x + 25 -6 ++6 x²y³ 9 y² 11. 36 x² 12 x(3 x + 3) + 36 x 9 y) 12. 81 x +9(9 x 8 y +

Factoring

factoring ¬To factor expressions find the GCF (greatest common factor) of the terms ¬Factoring is the opposite of distributing.

Find the GCF of each pair of monomials 1. 4 x, 12 x 12 cd, 36 cd 4 x 2. 18 a, 20 ab 2 a 3. 12 c d

Factor each expression 4. 12 a – 6 h 6(2 a – h) 7. 24 a – 4 4(6 a – 1) 5. 3 x + 9 12 x + y 3(x + 3) 8. 72 a + 9 n 8 a 9(8 a - 8 v + n) 6. Cannot be simplified 9. 8(a – v)

Solving equations

Steps to Solving Equations ¬ Equation: a mathematical sentence that uses an equal (=) sign. • Step 1: Get rid of the 10. Look at the sign in front of the 10, since it is subtraction we need to use the opposite operation (addition) to cancel out the 10 – Add 10 to both sides. Remember, what you do – 10 = you 50 have to do to to one side of the 2 n equation, +1 +1 the other. 0 0 2 n = 60

Steps to Solving Equations • Step 2: Next, we need to look at what else is happening to the variable. 2 n means that two is being multiplied to n, therefore we need to do the opposite (division) to “undo” the multiplication. – Divide both sides by 2. Remember, what you do to one side of the equation, you have to do to the other 2 n. = 60 2 2 n = 30

Steps to Solving Equations • Step 3: CHECK your solution!! First, rewrite the original equation – We already solved for n, so wherever you see the variable, n, plug in the answer. – Evaluate the equation, SHOWING ALL WORK! – Does it check? 2 n – 10 = 50 2 (30) – 10 = 60 – 50 10 = 50 50 = 50

Solve & Check 1. 105 = 10 n + 5 n = 10 2. n/5 + 3 = 6 n = 15 3. -44 + 7 n = 250 n = 42 4. -1/2 = -5/18 h h = -9/5 5. 200 = 100 – 25 n n = -4 6. -9. 4 + z = -3. 6 z = 5. 8

Solving equations practice

1. x – 3 = 19 2. a – 14 = 6 x = 22 a = 22 3. 9 x = 63 x = 22 5. 4. 5 x – 2 = 8 x = 22 6. 8 a + 5 = 53 x = -30 a = 22 7. -7 = c – 6 8. a – 3. 5 = 4. 9 c = 22 a = 22 8. 9. x – 2. 8 = 9. 5. x = 12. 3 10. 2. 25 + b = 1 b = 22 14. 2(b – 2) + b + 3

11. c = 1 3/7 13. m = 33/14 . 12. -8. 5 + r = -2. 1 r = 6. 4 14. 2(b – 2) + b = 6. 5 b = 2. 5

Solving multi-step equations

Steps to Solving Multi-Step Equations • Step 1: Distribute if necessary variable. – Distribute the 4 to the n and 5. 4(n – 5) - 7 = 9 + 2 n – 4 n 4 n – 20 - 7 = 9 + 2 n – 4 n

Steps to Solving Multi-Step Equations • Step 2: Combine like terms on each side of the equations. – On the left side -20 and -7 combine to get -27 – On the right side 2 n and -4 n combine to get -2 n 4 n – 20 - 7 = 9 + 2 n – 4 n 4 n – 27 = 9 – 2 n

Steps to Solving Multi-Step Equations • Step 3: Get all variables to one side of the equation. – First we want to get rid of the -27. Look at the sign in front of -27, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 27 to both 4 nsides – 27. = 9 – +22 n+2 7 7 4 n = 36 – 2 n

Steps to Solving multi-step Equations • Step 4: Get all “plain numbers” to one side of the equation – First we want to get rid of the -2 n. Look at the sign in front of -2 n, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 2 n to both 4 n sides. – 2 n = 36 +2 n 6 n = 36

Steps to Solving Multi-Step Equations • Step 5: Next, since we have all the variables on one side and all the “plain numbers” on the other side we need to look at what else is happening to the variable. – 6 n means the 6 is being multiplied by n, therefore we need to do the opposite (division) 6 n = 36 So, divide both to “undo” the multiplication. 6 6 sides by 6. n= 6

Steps to Solving Multi-Step Equations • Step 6: CHECK your solution!! First, rewrite the original equation – We already solved for n, so wherever you see the variable, n, plug in the answer. – Evaluate the equation, SHOWING ALL WORK! – Does it check? 4(n – 5) - 7 = 9 + 2 n – 4(6 – 5) - 4 n 7 = 9 + 2(6) – 4(1) - 7 =4(6) 9 + 12 – 24 4 – 7 = 21 -3 =24 -3

Solve & Check r = -2 1. 9 + 5 r = -17 – 8 r 2. 3(n + 5) + 2 = 26 n = 3 y = -11 3. 58 + 3 y = -4 y – 19 v = 12 4. 4 – 2(v – 6) = -8

Inequalities

Inequalities ¬Inequality: a mathematical sentence using <, >, ≥ , or ≤. – Example: 3 + y > 8. • Inequalities use symbols like < and > which means less than or greater than. • They also use the symbols ≤ and ≥ which means less than or equal to and greater than or equal to.

What’s the difference? • x < 4 means that x is less than 4 – 4 is not part of the solution – What number is in this solution set? • x ≤ 4 means that x can be less than OR equal to 4 – 4 IS part of the answer – What number is in this solution set?

You graph your inequalities on a number line: • This graph shows the inequality x < 4 • The open circle on 4 means that’s where the graph starts, but 4 is NOT part of the graph. • The shaded line and arrow represent all the numbers less than 4.

What is this inequality? X > -2

What is this inequality? X ≥ 2 1/2

Graphing inequality solution sets on a number line: • Use an open circle ( ) to graph inequalities with < or > signs. • Use a closed circle ( ) to graph inequalities with ≥ or ≤ signs.