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

Wallace made a quilt with the regular hexagon pattern shown. On each hexagon piece, he will sew a trim along each side. What expression represents the total length of trim on a hexagon?

Answer:
4 (5x + 2)

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

Build Understanding

Question 1.
A yard is shaped like an irregular quadrilateral with the side lengths shown.

A. Write an expression that shows the perimeter of the yard as the sum of the lengths of its sides.
Answer:
9\(\frac{1}{2}\) y + 9

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:

Question 2.
A path in a park forms the shape of an equilateral triangle with the side lengths shown.

A. Write an expression that represents the length of the path as a sum of the lengths of the three sides.
Answer:
(7.2d – 4) + (7.2d – 4) + (7.2d – 4)
21.6d -12.

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:

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

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).

D. Use the Distributive Property to expand the expression from Part C, and then simplify it.
3() = 3() + 3() =
Answer:
43.2d – 24

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:

Question 3.
Another path in the park goes around a square playground. The length of the entire path can be represented by the expression 20x + 8.

A. Find an expression that represents the length of each side of the square. Complete the expressions on the diagram.
Answer:
20x + 8

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

B. Complete the equivalent expression for the length of the path, 20x + 8.
20x + 8 = 4)
Answer:
4(5x + 2)

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

Question 4.
Apply what you have learned about adding, subtracting, factoring, and expanding linear expressions to simplify the expressions.
A. -.
(-t – 5) + (-2t + 3)
= -t + + + 3 Rewrite as addition of terms.
= t + Combine like terms
= –
Answer:
-3t – 2

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

B. Simplify (7 + 3d) – (5od – 5).
(7 + 3d) – (5d – 5)
= (7 + 3d) + ()(5d – 5)
= 7 + + + Rewrite as addition of terms.
= d + Combine like terms.
Answer:
2 – 2d

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

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

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

D. Expand -7(3x + 1).
-7(3x + 1)
= -7 () + (-7) () Use the Distributive Property
= – Simplify.
Answer:
-21x – 7

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

Question 5.
A regular polygon has sides that are all equal in length and angles that all have the same measure. A regular decagon has the side lengths as shown.

A. Model with Mathematics Write an expression for the perimeter of the regular decagon as a product of one side length. Explain.
Answer:
P = 110 ÷ 3

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 14.
Model with Mathematics The width of a rectangle is shown. The length is twice the width. Write an expression for the perimeter listing each side. Simplify the expression.

Answer:

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

Question 3.
Model with Mathematics The regular octagon has a perimeter represented by the expression shown. Write an expression to represent the length of one side of the octagon.

Answer:
6y – 5

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

Test Prep

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.