Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Signal-flow graph wikipedia , lookup
Quartic function wikipedia , lookup
Quadratic equation wikipedia , lookup
Cubic function wikipedia , lookup
System of polynomial equations wikipedia , lookup
Elementary algebra wikipedia , lookup
System of linear equations wikipedia , lookup
Solving One-Step Equations (2-2) Objective: Solve equations by using addition, subtraction, multiplication, and division. Solve Equations Using Addition or Subtraction O In an equation, the variable represents the O O O O number that satisfies the equation. To solve an equation means to find the value of the variable that makes the equation true. The process of solving an equation involves isolating the variable (with a coefficient of 1) on one side of the equation. Each step in this process results in equivalent equations. Equivalent equations have the same solution. Addition Property of Equality O If an equation is true and the same number is added to each side of the equation, the resulting equivalent equation is also true. O For any real numbers a, b, and c, if a = b, then a + c = b + c. 14 = 14 14 + 3 = 14 + 3 17 = 17 -3 = -3 +9 +9 6=6 Example 1 O Solve by adding. O Horizontal Method: h – 12 = -27 h – 12 + 12 = -27 + 12 h = -15 O Vertical Method: h – 12 = -27 +12 +12 h = -15 Check Your Solution: h – 12 = -27 -15 – 12 = -27 -27 = -27 Solve Equations Using Addition or Subtraction O Similar to the Addition Property of Equality, the Subtraction Property of Equality can also be used to solve equations. Subtraction Property of Equality O If an equation is true and the same number is subtracted from each side of the equation, the resulting equivalent equation is also true. O For any real numbers a, b, and c, if a = b, then a – c = b – c. 87 = 87 13 = 13 87 – 17 = 87 – 17 -28 -28 70 = 70 -15 = -15 Example 2 O Solve by subtracting. O Horizontal Method: c + 102 = 36 c + 102 – 102 = 36 – 102 c = -66 O Vertical Method: c + 102 = 36 -102 -102 c = -66 Check Your Solution: c + 102 = 36 -66 + 102 = 36 36 = 36 Solve Equations Using Multiplication or Division O In the equation x/3 = 9, the variable x is divided by 3. O To solve for x, undo the division by multiplying each side by 3. O This is an example of the Multiplication Property of Equality. Multiplication Property of Equality O If an equation is true and each side is multiplied by the same nonzero number, the resulting equation is equivalent. O For any real numbers a, b, and c, c ≠ 0, if a = b, then ac = bc. O If x = 5, then 3x = 15. Division Property of Equality O If an equation is true and each side is divided by the same nonzero number, the resulting equation is equivalent. O For any real numbers a, b, and c, c ≠ 0, if a = b, then a/c = b/c. O If x = -20, then x/5 = -20/5 or -4. O The reciprocal of a number can be used to solve equations. Example 3 O Solve each equation. 3 2 1 k 8 3 8 11 2 8 k 11 8 3 11 16 k 33 -75 = -15b -15 -15 5=b b=5 4 x 8•4 1 4 x = 32 Example 4 O We can also use reciprocals and properties of equality to solve real-world problems. O Ricardo is driving 780 miles to Memphis. He drove about 3/5 of the distance on the first day. About how many miles did Ricardo drive? 3 780 5 3 780 m 5 1 2340 m 5 m m = 468 miles Check Your Progress O Choose the best answer for the following. O Water flows through a hose at a rate of 5 gallons per minute. How many hours will it take to fill a 2400-gallon swimming pool? A. B. C. D. 4h 6h 8h 16 h m 2400 5 m = 480 minutes ÷ 60 Check Your Progress O Choose the best answer for the following. O Solve a – 24 = 16. Then check your solution. A. B. C. D. 40 -8 8 -40 a – 24 = 16 +24 +24 a = 40 a – 24 = 16 40 – 24 = 16 16 = 16 Check Your Progress O Choose the best answer for the following. O Solve 129 + k = -42. Then check your solution. A. B. C. D. 87 -171 171 -87 129 + k = -42 -129 -129 k = -171 129 + k = -42 129 + (-171) = -42 -42= -42 Check Your Progress O Choose the best answer for the following. O Solve 4 1 m 5 3 3 A. B. C. D. 23 1. 91 2 6. 7 2 1. 21 5 7 3 13 38 3 m 13 3 7 13 114 m 91 Check Your Progress O Choose the best answer for the following. O Solve 32 = -14c. -3 B. 46 C. 18 D. . 22 7 A. 32 = -14c -14 -14 16 c 7