Algebraic expressions

Question 1

a symbol that we use to represent a certain number.

Answer 1

Variable

Question 2

Example 2: Take a look at the expression x + y + z + 3. What are the variables in the expression?

Answer 2

Solution: x, y, and z are the variables used in the expression x + y + z.

x, y, and z represent certain quantities or numbers.

Since a variable represents certain quantities or values, this means that the value of a variable is not fixed. For instance, in x + y + z, the values of x, y, and z can be any number.

In the study of algebra, English letters are the most commonly used variables. Thus, in this reviewer, we will use letters to denote a variable that represents a certain value.

Question 3

a quantity with a fixed value.

Answer 3

cONSTANTS

Question 4

If you multiply a variable by a certain number, the latter is called a

Answer 4

numerical coefficients

Question 5

Example 1: Determine the numerical coefficient and literal coefficient in ¼ y.

Answer 5

Solution: The numerical coefficient is ¼ since it is the number multiplied by the variable y. Meanwhile, y is the literal coefficient since it is a variable multiplied by a number.

If a variable has no number written on its left, it means the numerical coefficient is 1. For instance, consider the variable x. Note that there is no number written on its left. This does not mean that it has no numerical coefficient. Instead, its numerical coefficient is 1. Thus, x can also be interpreted as 1x or “1 times x”.

However, in algebra, if the numerical coefficient is 1, we do not write it because it is already understood that a certain variable has a numerical coefficient of 1.

Question 6

Example 2: Determine the numerical coefficient of the following:
3y
0.23x
w

Answer 6

Solution: For item 1, the numerical coefficient is 3. For item 2, the numerical coefficient is 0.23. Lastly, for item 3, the numerical coefficient is 1.

Question 7

a mathematical expression that involves constants, variables, and arithmetic operations (addition, subtraction, multiplication, or division).

Answer 7

algebraic expressions

Question 8

Example 1: Determine the variables, constant, coefficient, and operations involved in 9 + 3xy – z.

Answer 8

Solution: The variables are x, y, and z. The constant is 9. Meanwhile, the operations involved are addition, multiplication (3xy can be interpreted as 3 times x times y), and subtraction. Furthermore, 3 is a numerical coefficient of xy.

Question 9

Example 2: Determine the variables, constants, and operations involved in the algebraic expression x⁄y – 2

Answer 9

Solution: Before we answer this, take note that in algebra, we usually indicate division as the ratio or a fraction between two numbers. Therefore, if we want to write x ÷ y, we write it as x/y instead.

Therefore, in x/y – 2, the variables are x and y, the constant is 2, and the operations involved are division and subtraction

Question 10

Example 1: Translate this verbal expression into an algebraic expression: “18 minus a number equals 5”.

Answer 10

Solution: The phrase “a number” means that we are not sure what that number is. This means that we need to represent it using a letter or symbol. In other words, we need to use a variable to represent that unknown number.

Let us use the letter g to represent this unknown number. Thus, if we translate “18 minus a number is 5” into an algebraic expression, we will obtain:

18 – g = 5

Question 11

The sum of 8 and a number

Answer 11

8 + x

Question 12

-6 plus a number

Answer 12

-6 + x

Question 13

A number increased by 7

Answer 13

x + 7

Question 14

3 more than a number

Answer 14

x + 3

Question 15

The total of a number and -10

Answer 15

x + (-10)

Question 16

Example 1: Translate “the sum of two numbers” into an algebraic expression.

Answer 16

Solution: The given sentence doesn’t explicitly state the values of two numbers. Thus, we need to use variables to represent them. Let us use the letters x and y to represent the numbers.

Since we have the keyword “sum”, it means that the numbers must be added.

Thus, the sentence can be translated as x + y

The answer is x + y

Question 17

Example 2:Translate “a number more than 18 is equal to 25” into an algebraic expression.

Answer 17

Solution:Let us assignkas the variable that represents the unknown number in the sentence.

It’s stated that“a number more than 18 is equal to 25”. Since the keyword “more than” is used, it means theoperation of additionis involved.

Again, the keyword “more than” implies that the first number mentioned in the sentence was added to the second number mentioned. This means that when we translate the sentence into an algebraic expression, we need to write the second number which is 18 as the first addend.

Thus, we can translate the sentence as 18 +k= 25.

The answer is 18 +k= 25.

Question 18

Thedifferencebetween a number and 15

Answer 18

x – 15

Question 19

9subtracted bya number

Answer 19

9-x

Question 20

9subtracted froma number

Answer 20

x-9

Question 21

15deducted bya number

Answer 21

15-x

Question 22

15deducted froma number

Answer 22

x – 15

Question 23

A numberdecreased by6

Answer 23

x – 6

Question 24

11minusa number

Answer 24

11 – x

Question 25

Example 1: Translate “A number subtracted from – 19 is equal to 5” into an algebraic expression.

Answer 25

Solution: Let us use the letter p as the variable that represents the unknown number. Recall that to translate a sentence with the keyword subtracted from into an algebraic expression, we are going to write it in the form – . Hence, the correct translation must be: – 19 – p = 5.

Question 26

Example 2: Translate “54 decreased by a number” into an algebraic expression

Answer 26

Solution: Let us use the letter q to represent the unknown number. The keyword “decreased by” implies that a number was subtracted from 54. Thus, the correct translation should be 54 – q

Question 27

The product of a number and 5

Answer 27

5x

Question 28

-3 multiplied by a number

Answer 28

-3x

Question 29

Twice of a number

Answer 29

2x

Question 30

Thrice of a number

Answer 30

3x

Question 31

½ of a number

Answer 31

½x

Question 32

7 times a number

Answer 32

7x

Question 33

Example 1: Translate “20% of a number is equal to 50” into an algebraic expression.

Answer 33

Solution: The keyword “of” indicates multiplication. Thus, the correct translation is 20%x = 50. You can also express the given percent into decimal. This means that 0.20x = 50 is also a translation for the given sentence.

Question 34

Example 2: Translate “The product of two numbers is equal to twice of another number” into an algebraic expression.

Answer 34

Solution: We have three unknown numbers involved. Thus, we need to use three letters as variables. Let us use the letters x, y, and z.  The product of two numbers can be translated as xy. Meanwhile, since the keyword twice indicates that a number is being doubled or multiplied by 2, then twice of another number can be translated as 2z. Hence, the correct translation of the given sentence should be xy = 2z.

Question 35

The quotient of 8 and a number

Answer 35

8 ÷ x or 8⁄x

Question 36

A number divided by 4

Answer 36

x ÷ 4 or x⁄4

Question 37

The ratio of a number and 2

Answer 37

x ÷ 2 or x⁄2

Question 38

A number is split equally into 3

Answer 38

x ÷ 3 or  x⁄3

Question 39

Example 1: What is “The sum of twice a number and 9” as an algebraic expression?

Answer 39

Solution: Let us start with the first keyword mentioned in the problem which is the sum. Recall that if the keyword sum is used in a sentence, it implies that there are numbers being added. The question now is: What are the numbers that are being added according to the given verbal expression? Let us read again the given verbal expression: The sum of twice a number and 9. It is clearly stated that there are two quantities that will be added–the quantity twice a number and 9.  We can now express our translation in this form for a while: twice a number + 9 Now, let us translate twice a number into an algebraic expression. Again, the word twice implies that a certain number is being doubled or multiplied by 2. Let us use x to represent the unknown number. Thus, twice of x is simply, 2x. Therefore, our final translation for The sum of twice a number and 9 is 2x + 9.

Question 40

Example 2: Translate “Twice the difference between two numbers” into an algebraic expression.

Answer 40

Solution: Let us start with the keyword twice. This keyword implies that a certain quantity or number will be multiplied by 2. What is this quantity that will be multiplied by 2 according to the given verbal expression? Let us read the given problem again: Twice the difference between two numbers. The statement tells us the difference between the two numbers is what will be multiplied by 2. Hence, we can translate it this way: 2(difference between two numbers). We use parenthesis to indicate multiplication. Now, let us translate the difference between two numbers into an algebraic expression. We have two unknown numbers so let us use x and y to represent them. Thus, we can translate the difference between two numbers as x – y. We replace the expression difference between two numbers in 2(difference between two numbers) with x – y. Hence, Twice the difference between two numbers is 2(x – y).

Question 41

Example 3: Write “the ratio of two numbers increased by 5” as an algebraic expression.

Answer 41

Solution: Let us start with the keyword ratio. The phrase the ratio of two numbers tells us that two numbers are involved in a division process. Let x and y be these two numbers. Hence, we can translate the ratio of two numbers as x ÷ y or x⁄y. For this problem, let us use x⁄y. The next keyword increased by in the ratio of two numbers increased by 5 tells us that the number 5 will be added to x⁄y. This means that we need to add 5 to our translation. Thus,   x⁄y+ 5 Therefore, if the ratio of two numbers increased by 5 will be written as an algebraic expression, you will have x⁄y+ 5

Question 42

The square root of a number  Cube root of a number

Answer 42

√x ∛x

Question 43

A number raised to 5 Square of a number Cube of a number

Answer 43

x5 x2 x3

Question 44

4 is equal to a number 3 plus a number yields 9 7 minus a number is 0

Answer 44

4 = x 3 + x = 9 7 – x = 0

Question 45

5 is not equal to a number 7 is greater than a number 7 is less than a number 0 is greater than or equal to a number 0 is less than or equal to a number 5 plus a number is at least 9 3 minus a number is at most 24 = x 3 + x = 9 7 – x = 0 0 ≥ x 0 ≤ x 5 + x ≥ 9

Answer 45

4 = x 3 + x = 9 7 – x = 0 0 ≥ x 0 ≤ x 5 + x ≥ 9 3 – x ≤ 2

Question 46

Example 1: Write “the sum of the square of a number and 3 is at least  9” as an algebraic expression.

Answer 46

Solution: The keyword sum tells us that certain quantities will be added. These quantities are the square of a number and 3. Let us use x to represent the unknown number. Its square can be represented as x2. Thus, the quantities that will be added are x2 and 3.  The sum of the square of a number and 3 is at least  9. This means that  x2  + 3 is greater than or equal to 9. Hence, the correct translation is x2.+ 3 ≥ 9

Question 47

Example 1: Lea has 150 books that she collected during her college years. She decided to give some of her books to her friends. After giving some to her friends, 40 books were left to Lea. Write an algebraic expression that will illustrate Lea’s scenario.

Answer 47

Solution: It’s stated in the given scenario that Lea has 150 books. She gave some of her books to her friends. 40 books were left to Lea after giving some of them. This can be interpreted as 150 books minus the number of books given equals 40. Let us use b to represent the number of books Lea gave to her friends. Thus, we have this algebraic expression: 150 – b = 40

Question 48

Example 2: A burger costs Php 32 each while a can of pineapple juice costs Php 25 each. Dario bought some burgers and cans of pineapple juice. Write an algebraic expression that shows how much Dario will pay for the burgers and cans of pineapple juice he bought.

Answer 48

Solution: It’s not specifically stated in the given scenario how many burgers and cans of pineapple juice Dario bought. Thus, we can use variables to represent the number of burgers and cans of pineapple juice he bought. Let b represent the number of burgers that Dario bought. Meanwhile, let p represent the number of cans of pineapple juice that Dario bought. Each burger costs Php 32. Thus, the total amount that Dario will pay for the burgers can be represented as 32b. Meanwhile, each can of pineapple juice costs Php 25. Thus, the total amount that Dario will pay for the cans of pineapple juice can be represented as 25p. Combining the total amount he will pay for the burgers and cans of pineapple juice: 32b + 25p. Thus, the answer is 32b + 25p.

Question 49

Example 1: Write x + 2 as a verbal expression.

Answer 49

Solution: The algebraic expression involves the operation of addition. Thus, you can use the keywords for addition in translating x + 2. Moreover, just use the words “a number” to translate the variable. Let’s use the keyword sum. We know that sum is written before the quantities that are added. Thus, one possible translation could be “the sum of a number and 2” Another possible translation is using the keyword “increased by”. You can translate x + 2 as “A number increased by 2”.

Question 50

Example 2: Translate 4z + 5 into a verbal expression.

Answer 50

Example 2: Translate 4z + 5 into a verbal expression.

Question 51

Example 3: Translate y2 ≤ 2 into a verbal expression.

Answer 51

Solution: y represents a certain number. Thus, we can translate it as “a number”. Meanwhile, y2 means that we squared that number. Hence, y2 can be translated as “the square of a number”. The symbol ≤ means less than or equal. We can use the phrase less than or equal for this symbol or the phrase at most. In this problem, let us use the phrase at most. Therefore, one possible translation for y2 ≤ 2 could be the square of a number is at most 2.

Question 52

Example 1: Evaluate x + 2 if x = 2

Answer 52

Solution: The variable in x + 2 is x. In this example, we have assigned a value to x which is x = 2. This means that we need to substitute or replace x with 2. Afterward, we perform the calculation to find the value of the algebraic expression when x = 2. Thus, x + 2 = 4 if x = 2.

Question 53

Example 2: Evaluate 5x + y if x = 2 and y = 0

Answer 53

Solution: We just substitute 2 for x and 0 for y in 5x + y. Take note that 5x is multiplication between 5 and x. Thus, once you substitute 2 for x, you will have 5(2) which implies “5 times 2”.  Hence, the answer is 10.

Question 54

Example 3: Evaluate 2x – 3(y + z) if x = 10, y = 1, and z = 3

Answer 54

Solution: Plug in the assigned values for x, y, and z in the given algebraic expression: Since, there are multiple operations involved, let use apply PEMDAS: Therefore, the answer is 8.

Question 55

Example 4: Evaluate 8a – 3b if a = ½ and b = – 2

Answer 55

Solution: Let us plug in the values of a and b to the given algebraic expression. Performing the operations involved: Thus, the answer is 10.

Question 56

1) Evaluate 2x 2y - 3xy + z at x = 1, y = 2, and z = - 1 a) - 1 b) - 2 c) - 3 d) - 4

Answer 56

C

Question 57

2) Write an algebraic expression that represents the phrase “the sum of a number and one-fourth of it”. a) x + 1 4 b) + x 1 4 1 4 c) x 1 4 d) x + x

Answer 57

D

Question 58

3) Translate into words. 3𝑥 2 a) The ratio of a number to thrice of it b) Three times a number decreased by 2 c) Thrice the ratio of a number to 2 d) The ratio of thrice a number to 2

Answer 58

D

Question 59

4) What is if a = 1? 𝑎 3 − 𝑎 2 a) - ⅔ b) - ⅓ c) ⅔ d) ⅓

Answer 59

A

Question 60

5) What is “The sum of two numbers is at least 5” in symbols? a) x + y < 5 b) x + y < 5 c) x + y > 5 d) x + y > 5

Answer 60

C