Transcript real world
LESSON 2.4 Variables and Equations Biology Lotus flowers like the one shown can be grown from seeds hundreds of years old. You will see how an equation can be used to estimate the year when an ancient lotus seed was formed. 1 LESSON 2.4 Variables and Equations An equation is a mathematical sentence formed by placing an equal sign, =, between two expressions. A solution of an equation with a variable is a number that produces a true statement when it is substituted for the variable. 2 LESSON 2.4 Variables and Equations EXAMPLE 1 Writing Verbal Sentences as Equations Verbal Sentence Equation The sum of x and 6 is 9. x+6=9 3 LESSON 2.4 Variables and Equations EXAMPLE 1 Writing Verbal Sentences as Equations Verbal Sentence Equation The sum of x and 6 is 9. x+6=9 The difference of 12 and y is 15. 12 – y = 15 4 LESSON 2.4 Variables and Equations EXAMPLE 1 Writing Verbal Sentences as Equations Verbal Sentence Equation The sum of x and 6 is 9. x+6=9 The difference of 12 and y is 15. 12 – y = 15 The product of –4 and p is 32. –4p = 32 5 LESSON 2.4 Variables and Equations EXAMPLE 1 Writing Verbal Sentences as Equations Verbal Sentence Equation The sum of x and 6 is 9. x+6=9 The difference of 12 and y is 15. 12 – y = 15 The product of –4 and p is 32. –4p = 32 The quotient of n and 2 is 9. n =9 2 6 LESSON 2.4 Variables and Equations EXAMPLE 2 Checking Possible Solutions Tell whether 9 or 7 is a solution of x – 5 = 2. Substitute 9 for x. x–5=2 ? 9–5 =2 4=2 ANSWER 9 is not a solution. 7 LESSON 2.4 Variables and Equations EXAMPLE 2 Checking Possible Solutions Tell whether 9 or 7 is a solution of x – 5 = 2. Substitute 9 for x. x–5=2 ? 9–5 =2 Substitute 7 for x. x–5=2 7–5 =2 ? 4=2 ANSWER 9 is not a solution. 2=2 ANSWER 7 is a solution. 8 LESSON 2.4 Variables and Equations Solving Equations Finding all solutions of an equation is called solving the equation. You can use mental math to solve a simple equation by thinking of the equation as a question. 9 LESSON 2.4 Variables and Equations EXAMPLE 3 Solving Equations Using Mental Math Equation Question x + 3 = 11 What number plus 3 equals 11? Solution 8 Check 8 + 3 = 11 10 LESSON 2.4 Variables and Equations EXAMPLE 3 Solving Equations Using Mental Math Equation Question Solution Check x + 3 = 11 What number plus 3 equals 11? 8 8 + 3 = 11 16 – m = 9 16 minus what number equals 9? 7 16 – 7 = 9 11 LESSON 2.4 Variables and Equations EXAMPLE 3 Solving Equations Using Mental Math Equation Question Solution Check x + 3 = 11 What number plus 3 equals 11? 8 8 + 3 = 11 16 – m = 9 16 minus what number equals 9? 7 16 – 7 = 9 20 = 5t 20 equals 5 times what number? 4 20 = 5(4) 12 LESSON 2.4 Variables and Equations EXAMPLE 3 Solving Equations Using Mental Math Equation Question Solution Check x + 3 = 11 What number plus 3 equals 11? 8 8 + 3 = 11 16 – m = 9 16 minus what number equals 9? 7 16 – 7 = 9 20 = 5t 20 equals 5 times what number? 4 20 = 5(4) y = –3 6 What number divided by 6 equals –3? –18 –18 = –3 6 13 LESSON 2.4 Variables and Equations EXAMPLE 4 Writing and Solving an Equation From 1998 to 2002, biologist Jane Shen-Miller grew several lotus plants from ancient seeds she found in China. The oldest seed was about 500 years old. Estimate the year when this seed was formed. SOLUTION First write a verbal model for this situation. Year seed was formed + Age of seed when it sprouted = Year seed sprouted 14 LESSON 2.4 Variables and Equations EXAMPLE 4 Writing and Solving an Equation From 1998 to 2002, biologist Jane Shen-Miller grew several lotus plants from ancient seeds she found in China. The oldest seed was about 500 years old. Estimate the year when this seed was formed. SOLUTION Let x represent the year when the seed was formed. Because you are only trying to estimate x (rather than determine x precisely), you can use 2000 for the year when the seed sprouted. This year simplifies mental calculations and lies within the given time period, 1998–2002. x + 500 = 2000 1500 + 500 = 2000 ANSWER Substitute for quantities in verbal model. Use mental math to solve for x. Because x = 1500, the seed was formed around the year 1500. 15