6.1 Order of Operations

Transcription

1 Chapter 6

2 6.1 Order of Operations is the study of unknown amounts. These unknown amounts are represented by. A is a specific amount added to or subtracted from a variable. For example, x + 7 or y 9. These examples are algebraic expressions because they are not set equal to anything. Other examples of expressions are 5x 6 and ½ x + ¾ y + 5. In these examples 5, ½ and ¾ are called coefficients. How are they different from constants?

3 When an expression is set equal to something it becomes an equation like 5x 6 = 9. This equation has one solution or there is one number that makes this statement true. An expression can equal anything, depending on what value the variable is given. 1. Evaluate the expression 3x + 5 for x = -3 and x = 7.

4 When an expression has many different operations more than one answer can be found depending on what operation is performed first. The Order of Operations insures that we perform the operations in the same order thereby getting the same answer. Order of Operations 1. Perform all operations in grouping symbols. Grouping symbols may be parentheses, brackets, braces or a fraction bar. 2. Perform all exponential operations. 3. Perform all multiplication and division from left to right. 4. Perform all addition and subtraction from left to right.

5 Use the Order of Operations to evaluate each expression. 1. 8(3) 4(2) = 2. 2³(4) = 3. 5(2⁵) 4(11 8)= 4. 7² + 9² 3² = 5. 4(10 6)³ (2 + 6) =

6 On page 315 do the following problems for practice: 9, 11, 14, 16, 28, 36 and 43. On pages do problems 10, 17, 18, 19, 20, 23, 25, 29, 32, 37, 39 and 44.

7 6.2 Linear Equations in One Variable Consider x + 2y 3x 4y + 9. What are the coefficients in the expression? What is the 9 called? Which terms are like terms? Simplify the expression.

8 When dealing with expressions we simplify the expression which means to combine like terms. 1. Simplify each expression or combine like terms. a. 5x + 3x b. 10y 3y c x + 9 2x d. -x + 6y y 4x + 2

9 When dealing with equations we solve the equation, finding the value of the variable that makes the statement true. Some properties of mathematics will help us solve equations. Addition/Subtraction Properties of Equality 5 3 = 5-3. Since 5 = 5, = and Example: Solve x 8 = 12 x = x = 20 Use the addition property of equality to isolate the x on the left side of the equal sign. Does x = 20 make the equation true? Example: Solve x + 7 = 15 What is being done to x? Undo it. x = 15 7 x = 8

10 2. Solve each equation. a x = -8 b x = 10 Multiplication/Division Property of Equality Since 5 = 5, 3 5 = 3 5 and x 5 Example: Solve = 8 Use the Multiplication Property to isolate x. x 5 5( ) = 5(8) x = 40

11 Example: Solve 3x = 15 What is being done to x? Undo it. 3x x = 5 3. Solve each equation. x x a. b. 16

12 On page 321 you will find a green box with steps to help you solve an equation with several operations in it. Looking at number 1 in that box, how might we solve the last equation -² ₅ x = 16? 4. Solve 2x Solve 4x 0.48 = 0.8x + 4

13 6. Solve 2x 9 = Solve 2 = 3 + 5(p+1) 8. Solve 3x + 4 = 5x + 6

14 9. Solve 10 = -5 +3(p 4) A is a comparison of two numbers. Ratios are read 3 to 5 and may be written 3/5 or 3:5. A is two ratios set equal to each other. To solve a proportion, cross multiply and solve the resulting equation. 10. Solve 5 x 7 11

15 11. Solve x Solve x 1 x Solve x 5 x 9 4 3

16 Examples 14 and 15 on pages 326 and 327 will be helpful with some word problems. On pages do problems 15, 16, 17, 19, 23, 25, 28, 34, 39, 40, 42, 45, 49, 51, 63, 65, 72, and 73.

17 6.3 Formulas Use the formula in each problem to find the value of the indicated variable for the values given. 1. A = bh Find A when b = 12 and h = V = 2πR²r² Find V when R = 3, r = 1 ¾, and π = E = mc² Find m when E = 400 and c = 4.

18 Next let s look at solving equations with both x and y. We will solve each equation for y. We have to remember how and when to combine like terms. We will also use the properties of equality from section 2 to solve for y. Solve for y. 4. 8x 4y = x + 7y = 15

19 6. 2x 3y + 52 = x + 3y 22 = 0 On pages do problems 7, 9, 19, 25, 30, 41, 43 and 49.

20 6.4 Applications of Linear Equations in One Variable Turn to page 340 in your textbook. Look at the phrases listed there and the matching mathematical expressions.

21 Turn to page 343. Complete each problem

22 On pages do problems 7, 9, 13, 19, 22, 26, 31, 32, 33, 34 and 39.

23 6.5 Variation A is an equation that relates one variable to one or more other variables through the operations of multiplication or division. In variation, the value of the two related variables increase together or decrease together. Direct Variation uses the formula y = kx and is read y varies as x or y is to x. K is a real number called the of proportionality. 1. y varies directly as x. Find y when x = 5 and k = 3.

24 2. H varies directly as L. If H = 15 when L = 50, find H when L = 10. Direct Variation as a Power uses the formula y = kxⁿ and is read y varies directly as nth power of x 3. Suppose y varies directly as the cube of x, and y = 24 when x = 2. Find y given x = 4.

25 Inverse Variation uses the formula y = inversely as x OR. k x and is read y varies y = k x n read y varies inversely as the nth power of x. 4. C varies inversely as J. If C = 7 when J = 0.7, find C when J = 12.

26 5. The current in a simple electrical circuit varies inversely as the resistance. If the current is 80 amps when the resistance is 10 ohms, find the current if the resistance is 16 ohms. 6. Suppose p varies inversely as the cube of q and p = 100 when q = 3. Find p, given q = 5.

27 Joint Variation uses the formula y = kxz and is read y varies jointly as x and z. 7. F varies jointly as D and E. Find F when D = 3, E = 10 and k = If x varies jointly as y and z², and x = 231 when y = 3 and z = 2, find x when y = 5 and z = 4.

28 9. Z varies jointly as W and Y. If Z = 12 when W = 9 and Y = 4, find Z when W = 50 and Y = 6.

29 Combined Variations are a combination of the variations used above. 10. Suppose z varies jointly as x and y² and inversely as w. Also, z = 3/8 when x = 2, y = 3 and w = 12. Find z, given x = 4, y = 1 and w = 6. z varies jointly as x and y² z = z varies inversely as w z = Put the two together: z =

30 11. t varies directly as the square of d and inversely as f. If t = 192 when d = 8 and f = 4, find t when d = 10 and f = 6. On pages do problems 23, 29, 31, 33, 34, 35, 36, 38, 42, 43, 46 and 49.