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Glencoe Math Course 2, Grade: 7 Publisher: Glencoe/McGraw-Hill

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Into Math Grade 7 Module 7 Lesson 2 Answer Key Add, Subtract, and Factor Linear Expressions with Rational Coefficients

We included  HMH Into Math Grade 7 Answer Key  PDF   Module 7 Lesson 2 Add, Subtract, and Factor Linear Expressions with Rational Coefficients to make students experts in learning maths.

HMH Into Math Grade 7 Module 7 Lesson 2 Answer Key Add, Subtract, and Factor Linear Expressions with Rational Coefficients

I Can identify problems that require me to add, subtract, factor, and expand linear expressions with rational coefficients.

Spark Your Learning

Turn and Talk Which is a like term with 5x: 10x, 5y, or 5? Explain. Answer: 5x + 10x = 5y + 5

Build Understanding

Explanation: An expression that shows the perimeter of the yard as the sum of the lengths of its sides is 9\(\frac{1}{2}\) y + 9

B. Rewrite the expression as an addition of terms. Answer:

C. Use the Commutative Property of Addition and the Associative Property of Addition to rearrange the terms so that all of the y-terms are together and all of the integer terms are together. Answer:

D. How can you identify like terms in an expression? Answer:

E. Simplify the expression by combining like terms. Answer:

F. Show how you used the Distributive Property to combine the y-terms. Answer:

Turn and Talk Explain why you might want to combine like terms before evaluating an expression. Discuss why the expressions in Part A and Part E look so different. Answer:

Explanation: Given The side of the equilateral traingle is (7.2d – 4). The expression that represents the length of the path as sum of the lengths of the three sides are+ (7.2d – 4) + (7.2d – 4) + (7.2d – 4) 21.6d -12.

B. Use the Commutative and Associative Properties of Addition to reorder and group like terms in your expression from Part A. Then combine like terms to simplify the expression. Answer:

Explanation: an equivalent expression for the length of the path using the factor 3 and the length of one side is 3(7.2d – 4).

Explanation: Let us solve the given equation. 3(7.2d – 4) = 3(7.2d – 4) + 3(7.2d – 4) (3 × 7.2d) – (3 × 4) + (3 × 7.2d) – (3 × 4) 21.6d – 12 + 21.6d -12 43.2d – 24

Turn and Talk What do you notice about your expressions in Part B and Part D? Discuss your observations. Answer:

Explanation: An expression that represents the length of each side of the square is 20x + 8.

Explanation: The equivalent expression for the length of the path is 4(5x + 2).

Turn and Talk In Part B, you factored the expression 20x + 8 by writing it as a product. Explain how you used division and the Distributive Property to do this. Answer: The division and distributive property is A × ( B × C).

Step It Out

Explanation: Let us calculate the given expression. (-t – 5) + (-2t + 3) -t – 5 -2t + 3 -3t – 2

Explanation: Let us calculate the given expression. Simplify (7 + 3d) – (5d – 5). 7 – 5 + 3d – 5d 2 – 2d

Explanation: The Factor 30x – 5 using the greatest common factor (GCF) is 5(6x – 1).

Explanation: Expand -7(3x + 1). -7(3x + 1) -7 × 3x – 7 × 1 the Distributive Property -21x – 7

Check Understanding

Question 1. Simplify the expression 4x – 6x + 15 – x – 4. Answer: -3x + 11

Explanation: Let us solve the given expression The expression is 4x – 6x +15 – x -4 4x -6x – x + 15 – 4 -3x + 11

Question 2. Use the Distributive Property to expand the expression 3(3x + 6). Then simplify the expression. Answer: 9x + 18

Explanation: Apply Distributive law 3(3x + 6) 3 × 3x + 3 × 6 9x + 18

Question 3. Factor 24x – 20 using the GCF. Answer: 4 (6x – 5)

Explanation: Factor the given expression 4(6x -5)

Question 4. Simplify the expression 4(\(\frac{3}{2}\)f + 1) – 7(f + 5). Answer: -f + 39

Explanation: The expression 4 × \(\frac{3}{2}\)f + 4 – 7f + 35 6f + 4 – 7f + 35 -f + 39

On Your Own

Explanation: Given, a = 5 – 1\(\frac{1}{3}\)t a = 5 – \(\frac{4}{3}\)t a = (15 – 4) ÷ 3 a = 11 ÷ 3 Perimeter = 10 × a P = 10 × \(\frac{11}{3}\) P = 110 ÷ 3

B. Use the Distributive Property to expand the expression from Part A. Then simplify. Answer:

Question 6. Factor 14f + 21 using the GCF. Answer:

Question 7. Model with Mathematics A pentagon has side lengths: (12 + 4x), (10 + 8x), (15 + 3x), (9 + 2x), and (14 + 3x). Write a simplified expression that represents the perimeter of the pentagon. Explain. Answer:

For Problems 8-9, factor the expressions using 3 as one factor.

Question 8. 3x – 30 Answer: x = 10

Explanation: Let us solve the given expression 3x – 30 x = 30 ÷ 10 x = 3

Question 9. 3x + 15 Answer: x = -5

Explanation: Let us solve the given expression 3x + 15 = 0 3x = -15 x = \(\frac{-15}{3}\) x = -5

For Problems 10-11, simplify the expressions using properties of operations.

Question 10. (4x – 7.2) + (-5.3x – 8) Answer: -1.3x – 15.2

Explanation: Let us solve the given expression (4x – 7.2) + (-5.3x – 8) -1.3x – 15.2

Question 11. (t – 1) – (-7t + 2) Answer:

For Problems 12-13, expand the expressions using the Distributive Property. Then simplify the expressions.

Question 12. 4(7x + 3) Answer: 28x + 12

Explanation: Let us solve the given expression 4(7x + 3) 28x + 12

Question 13. 9(3y – 5) Answer: 27y – 45

Explanation: Let us solve the given expression 9(3y – 5) 27y – 45

Question 15. Katelyn drew a pentagon. The side lengths are (6.7t + 4.3), (-t + 11), (4.8t + 3), (9.7t – 0.4), and (8.6t – 0.2). A. Model with Mathematics Write an expression for the perimeter. Group the like terms. Then combine like terms to simplify. Answer: 28.8t – 17.7

Explanation: An expression for the perimeter is (6.7t + 4.3) + (-t + 11) + (4.8t + 3) + (9.7t – 0.4) + (8.6t – 0.2) 6.7t – t + 4.8t + 9.7t + 8.6t + 4.3 +11 + 3 – 0.4 – 0.2 28.8t – 17.7

B. What two properties allowed you to reorder and regroup the terms? Answer:

For Problems 16-17, simplify the expressions using properties of operations.

Question 16. (-6s – 7\(\frac{2}{5}\)) + (-6s + 6) Answer: (-6s – \(\frac{37}{5}\)) + (-6s + 6) ((-30s – 37) + 5 (-6s +6)) ÷ 5 (-30s -37 – 30s + 30) ÷ 5 (-60s + 7) ÷ 5

Question 17. 5(y – 7) – (2y + 9) Answer: 3y – 26

Explanation: Let us solve the given expression 5(y -7) – (2y + 9) 5y – 35 – 2y + 9 3y – 26

For Problems 18-19, expand and simplify the expressions using properties of operations.

Question 18. 8(3x – 7) Answer: 24x – 56

Explanation: Let us solve the given expression 8(3x – 7) 24x – 56

Question 19. 14(3b + 2) Answer: 42b + 28

Explanation: Let us solve the expression. 14(3b + 2) 42b + 28

In Problems 20-21, simplify using properties of operations and then factor the expressions using the GCF.

Question 20. (10p + 10) + (8p – 1) Answer: 18p – 9

Explanation: Let us solve the given expression (10p + 10) + (8p – 1) 10p + 8p + 10 – 1 18p + 9

Question 21. (2g + 2) – (-4g – 7) Answer: 6g + 18

Explanation: Let us solve the given expression (2g + 2) – (-4g – 7) 2g + 4 + 4g + 14 6g + 18

I’m in a Learning Mindset!

How can I apply my prior knowledge of the Distributive Property to factoring algebraic expressions? Answer:

Lesson 7.2 More Practice/Homework

Question 1. Model with Mathematics Write a simplified expression that represents the perimeter of an irregular quadrilateral with side lengths (2\(\frac{1}{4}\)t – 5). (4t + 3), (\(\frac{1}{2}\)t – 1), and (3t + 2). Answer: (29t – 16) ÷ 4

Explanation: Given, a = (2\(\frac{1}{4}\)t – 5) b = (4t + 3) c = (\(\frac{1}{2}\)t – 1) d = (3t + 2) Perimeter = (2\(\frac{1}{4}\)t – 5) + (4t + 3) + (\(\frac{1}{2}\)t – 1) = (\(\frac{9}{4}\)t – 5) + (4t + 3) + (\(\frac{1}{2}\)t – 1) = (9t – 20) ÷ 4 + (4t + 3) + (t – 2) ÷ 2 = ((9t – 20) + 4 (4t + 3) + 4(t -2)) ÷ 4 = (9t – 20 + 16t + 12 + 4t – 8) ÷ 4 = (29t – 16) ÷ 4

Question 2. Reason The length of a rectangle is represented by 4 + 6x. The width is half the length. What expression represents the perimeter of the rectangle? Explain your reasoning. Answer: 12 + 18x

Explanation: l = 4 + 6x w = 2 + 3x Perimeter of the rectangle is 2 ( 4 + 6x + 2 + 3x) = 2 ( 6 + 9x) = 12 + 18x

Explanation: Let us solve the given expression An expression to represent the length of one side of the octagon. Given perimeter = 48y – 40 Octagen as 8 sides (48y – 40) ÷ 8 6y – 5.

For Problems 4-9, simplify the expressions using properties of operations.

Question 4. Math on the Spot 5(x – 4) + 2x Answer: 7x – 20

Explanation: Let us solve the given expression 5(x – 4) + 2x 5x – 20 + 2x 7x – 20

Question 5. 18t – 3 – 5t + 8 Answer: 15t + 5

Explanation: Let us solve the given expression 18t – 3 – 5t + 8 15t + 5

Question 6. 7.5 + 5f + 16.2 + 2f Answer: 23.7 + 7f

Explanation: Let us solve the given expression 7.5 + 5f + 16.2 + 2f 23.7 + 7f

Question 7. -8(1 + x) + 7x Answer: -8 + 6x

Explanation: Let us solve the given expression. -8 – x + 7x -8 + 6x

Question 8. 7\(\frac{1}{3}\)t – (10\(\frac{2}{3}\)t – 6) Answer: 4 ÷ 3

Explanation: Let us solve the given expression 7\(\frac{1}{3}\)t – (10\(\frac{2}{3}\)t – 6) \(\frac{22}{3}\)t – \(\frac{32}{3}\) – 6 –\(\frac{10}{3}\) – 6 (-10 + 18) ÷ 6 8 ÷ 6 4 ÷ 3

Question 9. (-r – 5) – (-2r – 4) Answer: r – 1

Explanation: Let us solve the given expression (-r -5) – (-2r -4) -r – 5 + 2r + 4 r – 1

For Problems 10-11, expand and simplify the expressions using properties of operations.

Question 10. 7(11c + 3) Answer: 77c + 21

Explanation: Let us calculate the given expression. 7(11c + 3) (7 × 11c) + (7 × 3) 77c + 21

Question 11. 6(7y – 8) Answer: 42y – 48

Explanation: Let us calculate the given expression 6(7y – 8) (6 × 7y) – (6 × 8) 42y – 48

Question 12. A square has a perimeter represented by the expression 8. 8s – 20. Write an expression to represent the length of one side of the square. Answer: -2.2s – 5

Explanation: Given, A square with perimeter expression -8.8s – 20. To find expression for length of one side of the square. Length of the square is perimeter ÷ 4 -8.8s – 20 ÷ 4 -2.2s – 5 Hence the correct answer is -2.2s – 5.

Question 13. Simplify -5(7 + x) + 2\(\frac{5}{6}\)x. Answer: -35 -5x + \(\frac{17}{6}\)x -35 – 5x + \(\frac{17}{6}\)x (-210 – 30x + 17x) ÷ 6 (-210 – 13x) ÷ 6

Question 14. Which expression is equivalent to 9y + 2(1 – 5y)? (A) 4y + 2 (B) 19y + 2 (C) y + 2 (D) -y + 2 Answer: -y + 2

Explanation: Let us solve the given expression The given expression is 9y + 2(1 – 5y). 9y + 2(1 – 5y) 9y + 2 – 10y -y + 2

Question 15. An irregular pentagon has side lengths of (x + 3), (2x – 4), (4x + 5), (3x – 1), and x. Which simplified expression represents the pentagon’s perimeter? (A) 11x – 3 (B) 24x + 60 (C) 11x + 3 (D) -9x + 3 Answer: 11x + 3

Explanation: Let us calculate the given expression The side length of pentagon is (x + 3), (2x – 4), (4x + 5), (3x – 1), x (x+3) + (2x-4) + (4x + 5) + (3x – 1) + x x + 2x + 4x + 3x + x + 3 -4 + 5 -1 11x + 3

Spiral Review

Question 16. Jovan is 15 years old. His sister is 6 years older than \(\frac{1}{3}\) his age. How old is Jovan’s sister? Answer: 11

Explanation: Jovan is 15 years His sister is 6 years older than \(\frac{1}{3}\) 6 + \(\frac{1}{3}\) × 15 6 + 5 11 Jovan’s sister is 11 years old.

Question 17. Steven multiplies all the integers from -99 to -90, including -99 and -90. Should his answer be positive or negative? Explain your thinking. Answer: 8910

Explanation: Based on the given question, calculate -99 × -90 Now determine the sign for multiplication or division 99 × 90 8910.

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NAME DATE PERIOD Lesson 8 Extra Practice Factor Linear Expressions Find the GCF of each pair of monomials. 1. 28, 42x 14 2. 21a, 45 3 3. 16m, 56n 8 4. 42x, 56y 14 5. 20c, 28cd 4c 6. 5ab, 6b b 7. 7x,.

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Lesson 4 homework practice powers of monomials FAQ

How do you factor expressions step by step.

7:44 11:53 How To Factor Polynomials The Easy Way! - YouTube YouTube Start of suggested clip End of suggested clip Track. So now we're going to factor out x minus 3.. If we take out x minus 3 from this term we'reMoreTrack. So now we're going to factor out x minus 3.. If we take out x minus 3 from this term we're left with 2x. If we take out x minus 3 from that term we're left with plus 1.

How do you factor a linear equation?

To factor a linear equation, we can use this property and the following steps: Identify the greatest common factor of m and b, call it a. If there is no common factor, then the linear equation cannot be factored. Rewrite m and b in terms of a. Use the distributive property in reverse to factor a out of mx + b.

How do you factor linear expressions?

1:21 5:30 Factoring Linear Expressions - YouTube YouTube Start of suggested clip End of suggested clip And because we have a plus sign here up across the middle. And you say 5 times what gives us 10 X.MoreAnd because we have a plus sign here up across the middle. And you say 5 times what gives us 10 X. So 5 times 2x that will give us 10 X. And then we say 5 times what gives us 15 and that's 3.

What is the first step in factoring the linear expression?

To factor an algebraic linear expression: Find the greatest common factor (GCF). Factor out that value by dividing. The distributive property in reverse shows the factoring of an expression.

How do you solve linear expressions by factoring?

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