EXPRESSIONS AND EQUATIONS

LESSON 4.1

P

atty’s mother owns a flower shop. On Mother’s Day, Patty helps her mother deliver flowers. Her mother pays her $10 for the day plus an additional $2 for each delivery she makes. The expression that represents the total amount of money which Patty has made is: 10 + 2d A variable is a symbol that represents one or more numbers. In this expression, d represents the number of deliveries Patty has made. An algebraic expression is a mathematical expression that contains numbers, operations (such as add, subtract, multiply or divide) and variables. You can evaluate an algebraic expression by substituting a number for the variable to find its value. Some algebraic expressions will have more than one variable. If this is the case, you will substitute each variable with a given number. After substituting each variable with a number, you will use the order of operations to find the value of the expression. The value of an algebraic expression changes depending on the value of the variable. If Patty made twelve deliveries on Mother’s Day, you can evaluate the expression by substituting 12 for d. 10 + 2(12) = $34 Patti made $34 delivering flowers.

EXAMPLE 1

Evaluate each algebraic expression. a. 5x − 3 when x = 8 b. (y + 2)² + p when y = −7 and p = 9

Solutions

a. Write the expression. 5x − 3 Substitute 8 for x. 5(8) − 3 Multiply. 40 − 3 Subtract. 37 b. Write the expression. (y + 2)² + p Substitute −7 for y and 9 for p. (−7 + 2)² + 9 Add inside parentheses. (−5)² + 9 Square −5. 25 + 9 Add. 34

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Lesson 4.1 ~ Expressions and Equations

When you write algebraic expressions, it is necessary to know key words that represent the four basic operations in mathematics. Addition

Subtraction

Multiplication

Division

sum increased by more than plus

difference decreased by less than minus

product multiplied by times of

quotient divided by

EXAMPLE 2

Write an algebraic expression for each phrase. a. the product of seven and m b. two more than six times x c. f divided by three then decreased by four

Solutions

a. “product” means multiplication b. “more than” means addition “times” means multiplication

7m 2 + 6x or 6x + 2

f c. “divided by” means division ​ __  ​− 4 3 “decreased by” means subtraction

Once you are able to write and evaluate expressions, you are able to work with equations. An equation is a mathematical sentence that contains an equals sign (=) between two expressions. Equations that involve a variable are neither true nor false until the equation is evaluated with a given value for the variable. A value is considered the solution of an equation if it makes the equation true.

x − 6 = 13





19 − 6 = 13 13 = 13

To test if a value makes an equation true, substitute the value for the variable. If one side of the equation is equal to the other side, the value is a solution to the equation.



Lesson 4.1 ~ Expressions and Equations 

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EXAMPLE 3

Determine if the number given is the solution of the equation. a. −2y + 6 = −16 Is 5 the solution? x __ b. ​   ​− 4 = 4 Is 40 the solution? 5

Solutions

a. See if the given value makes the equation true. Write the equation. Substitute 5 for y. Multiply. Add. 5 is NOT the solution. b. Write the equation.

−2y + 6 = −16 −2(5) + 6 =? −16 −10 + 6 =? −16 −4 ≠ −16

__ ​ x ​− 4 = 4

5

40  ​− 4 =? 4 ___ Substitute 40 for x. ​  5 Divide.  8 − 4 =? 4

Subtract. 40 IS the solution.

4 = 4

EXERCISES Write an algebraic expression for each phrase. 1. the sum of y and five 2. nine times m then increased by six 3. six less than the product of z and seven 4. the quotient of c and three 5. ten more than twice x 6. one subtracted from p Write a phrase for each algebraic expression. 7. y − 2 8. 4d − 7 9. 60 + 2x 10. Write three different phrases for 3x + 2. Evaluate each expression. 11. x − 5 when x = 24 12. 12b + 3 when b = −4

14. 50 − 7k when k = 10

15. _​  12 ​m   + _​ 34 ​ when m = _​ 14 ​



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Lesson 4.1 ~ Expressions and Equations

13. y² when y = 5 16. 1.3h + 3.7 when h = 5

Evaluate each expression.

17. −3d − 2c when d = 2 and c = 10

18. __ ​  2v  ​− 3w when v = 40 and w = 10 5

19. Carol Middle School is planning a special “Movie Day” for all students who had no





missing assignments last quarter. The principal will purchase enough containers of popcorn and juice for all the students that will be attending. The popcorn costs $2.50 per container. Bottles of juice cost $3.00 each. The principal uses the algebraic expression 2.50x + 3.00y to calculate the total expenses. a. What does the x variable represent? b. What does the y variable represent? c. The principal bought 8 containers of popcorn and 11 bottles of juice. How much did she spend?

20. The baseball team sold boxes of oranges and holiday wreaths as a

fundraiser. The team made a profit of $6.25 for each box of oranges sold. The team made $8.00 in profit for each wreath sold. a. Let b represent a box of oranges and w represent a holiday wreath. Write an expression that represents the total profit made based on the number of boxes of oranges and wreaths that are sold. b. The team sold 40 boxes of oranges and 25 holiday wreaths. What was their total profit? Determine if the number given is the solution of the equation. 21. 2x − 6 = 3 Is 5 the solution? 22. −5y − 4 = −19 23. __ ​  m  ​+ 10 = 12  ​ _12 ​ Is 20 the solution? 24. _​  23 ​h − 2 = −2 8

Is 3 the solution?

25. 2(x + 4)2 = 19

Is 1.5 the solution?

Is −1 the solution?

26. −2 + 10x = 13

Is −6 the solution?

Write an algebraic equation for each phrase. Use x as the variable for the missing number. Match each equation to its solution.

27. a number plus six equals ten

A. x = 21

28. three times a number then decreased by fourteen equals four

B. x = 6

29. a number divided by six then increased by four is nine

C. x = 7

30. a number squared is forty-nine

D. x = 30

31. nine more than half of a number is fourteen

E. x = 4

32. one more than the quotient of a number and three is eight

F. x = 10



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REVIEW Find each sum, difference, product or quotient. 33. 2  ​ _14 ​ (​ −5  ​ _13 ​  )​ 34. _​  45 ​+ ​ − ​ _12 ​  ​ 35. 1 ​ _12 ​ − 3  ​ _34 ​

(  )

36. − ​ _89 ​÷ (​ −2 )​

(  )

9  ​  ​ 37. 5 − (​ −1  ​ _27 ​  )​ 38. __ ​  5  ​ − ​ __ 27 10

T ic -T ac -T oe ~ BMI Body Mass Index (BMI) is a number calculated from a person’s weight and height. BMI is a reliable indicator of body fat for adults. BMI is an inexpensive and easy-to-perform method of screening for weight categories that may lead to health problems. BMI is calculated using the formula: weight ∙ 703 ________________ ​       (height in inches)²  ​ BMI scores can be used to categorize adults into standard weight status categories: BMI

Weight Status

Below 18.5

Underweight

18.5 – 24.9

Normal

25.0 – 29.9

Overweight

30.0 and Above

Obese

Calculate the BMI for each individual and classify them using the weight status categories above. Show all work.

1. Female, 5’ 5”, 140 pounds

2. Male, 6’ 1”, 275 pounds

3. Male, 5’ 10”, 180 pounds

4. Female, 5’ 7”, 115 pounds

Write an equation and solve it to find the weight of a person with the given height and BMI. Show all work and round the answer to the nearest tenth of a pound.

5. Male, 5’ 8”, BMI = 22.2

6. Female, 5’ 1”, BMI = 30

7. Male, 6’ 4”, BMI = 27.4

8. Female, 5’ 9”, BMI = 18.3

9. Research obesity to find at least three diseases that are caused by obesity. 10. Write out at least two ways people can help keep their weight status in the normal BMI range. 112 

Lesson 4.1 ~ Expressions and Equations