Chapter 1 Algebra: Understand Numerical And Algebraic Expressions
Guided Practice 5
Page 39 Exercise 1 Answer
We can use Commutative Property of Addition to write an equivalent expression for y + \(\frac{1}{2}\)
\(y+\frac{1}{2}=\frac{1}{2}+y\)Using, Commutative Property of Addition: a + b = b + a
Result
Commutative Property of Addition
The equivalent expression is \(\frac{1}{2}\) + y
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Page 39 Exercise 2 Answer
Let us substitute the value of z to find \(z^3\) and 3z are equivalent or not.
For algebraic expression to be equivalent they must have the same value for any number substitute for the same variable.
The expression \(z^3\) and 3z are not equivalent when z = 1 and z = 2
So, \(z^3\) and 3z are not equivalent expressions.
Result
\(z^3\) and 3z are not equivalent expressions.
Page 39 Exercise 3 Answer
2(r + 3) Evaluate
= 2(r) + 2(3) Using Distributive Property
= 2r + 6 Multiply
Result
2r + 6
Page 39 Exercise 4 Answer
6(4s − 1) Evaluate
= 6(4s) − 6(1) Using Distributive Property
= (6 ⋅ 4)s − 6(1) Using Associative Property of Multiplication
= 24s − 6 Multiply
Result
24s − 6
Page 39 Exercise 5 Answer
8t + 2 Evaluate
= 2(4t) + 2(1) Using Distributive Property
= 2(4t + 1) 2 is a common factor
Result
2(4t + 1)
Page 39 Exercise 6 Answer
3(m + 3) Evaluate
= 3(m) + 3(3) Using Distributive Property
= 3m + 9 Multiply
Result
3m + 9
Page 39 Exercise 7 Answer
20n − 4m Evaluate
= 4(5n) − 4(m) Using Distributive Property
= 4(5n − 4) 4 is the common factor
Result
4(5n − 4)
Page 39 Exercise 8 Answer
\(4\left(3 p+2 \frac{1}{2}\right)\) Evaluate
= \(4(3 p)+4\left(2 \frac{1}{2}\right)\) Using Distributive Property
= \((4 \cdot 3) p+4\left(\frac{5}{2}\right)\) Using Associative Property of Multiplication
= 12p + 10 Multiply
Result
12p + 10
Page 39 Exercise 9 Answer
3(x − 6) Evaluate
= 3(x) − 3(6) Using Distributive Property
= 3x − 18 Multiply
Result
3(x − 6) and 3x − 18 are equivalent expressions
Page 39 Exercise 10 Answer
2x + 10 Evaluate
= 2(x) + 2(5) Using Distributive Property
= 2(x + 5) 2 is the common factor
Result
2x + 10 and 2(x + 5) are equivalent expression.
Page 39 Exercise 11 Answer
\(8\left(2 y+\frac{1}{4}\right)\) Evaluate
= \(8(2 y)+8\left(\frac{1}{4}\right)\) Using Distributive Property
= 16y + 2 Multiply
Result
16y + 2 are equivalent expression
Page 39 Exercise 12 Answer
5.7 + (3z + 0.3) Equivalent
= 5.7 + 3z + 0.3 Open parentheses
= 5.7 + 0.3 + 3z Group like terms
= 3z + 6 Combine like terms
= 3(z) + 3(2) Using Distributive Property
= 3(z + 2) 3 is the common factor
Result
3(z + 2) is the equivalent expression
Page 39 Exercise 13 Answer
5w − 15 Equivalent
= 5(w) − 5(3) Using Distributive Property
= 5(w − 5) 5 is the common factor
Result
5(w − 5) is the equivalent expression
Page 39 Exercise 14 Answer
2x + 4y Equivalent
= 2(x) + 2(2y) Using Distributive Property
= 2(x + 2y) 2 is the common factor
Result
2(x + 2y) is the equivalent expression
Page 39 Exercise 15 Answer
10(\(y^2\) + 2.45) Equivalent
= \(10\left(y^2\right)+10(2.45)\) Using Distributive Property
10\(y^2\) + 24.5 Multiply
Result
10\(y^2\) + 24.5 is the equivalent expression
Page 39 Exercise 16 Answer
\(\frac{3}{4} \cdot\left(z^3 \cdot 4\right)\) Equivalent
= \(\frac{3}{4} \cdot z^3 \cdot 4\) Open Parentheses
= 3\(z^3\) Multiply
Result
3\(z^3\) is the equivalent expression
Page 40 Exercise 17 Answer
Length = 5 and Width = 2x − 1
Algebraic expression for the area of rectangle:
Length × Width = 5 ⋅ (2x − 1)
Result
5 ⋅ (2x − 1)
Page 40 Exercise 18 Answer
5(2x − 1) Evaluate
= 5(2x) − 5(1) Using Distributive Property
= 10x − 5 Multiply
Result
10x − 5 is the equivalent expression
Page 40 Exercise 19 Answer
10x − 5 Evaluate
= \(10\left(5 \frac{1}{2}\right)-5\) Substitute x = \(5 \frac{1}{2}\)
= \(10 \cdot \frac{11}{2}-5\) Open parentheses
= 55 − 5 Multiply
= 50 Subtract
Result
Area = 50 square units
Page 40 Exercise 20 Answer
Number of magnifying glasses . Cost of each magnifying glass + Number of safety glasses ⋅ Cost of each safety glass Given
7 ⋅ 1.25 + 7 ⋅ 3.75 Evaluate
= 8.75 + 26.25 Multiply
= $35 Add
Result
The total cost is $35
Page 40 Exercise 21 Answer
5y − 20 Evaluate
= 5(y) − 5(4) Using Distributive Property
= 5(y − 4) 5 is the common factor
Result
5 is the common factor in the expression 5y − 20
Page 40 Exercise 22 Answer
2(2n − 1) Evaluate
= 2(2n) − 2(1) Using Distributive Property
= 4n − 2 Multiply
Result
Yes, Chrish is correct that the expression 4n − 2 and 2(2n − 1) are equivalent
Page 40 Exercise 23 Answer
(f ⋅ \(g^2\)) + 5 − (\(g^2\) ⋅ f) Evaluate
= f\(g^2\) + 5 − f\(g^2\) Multiply
= 5 Simplify
Result
5 is the only one term and is equivalent to the expression (f ⋅ \(g^2\)) + 5 − (\(g^2\) ⋅ f)
Page 40 Exercise 24 Answer
8.5 + (2s + 0.5)
= (8.5 + 2s) + 0.5 Using Associative Property of Addition
8.5 + (2s + 0.5)
= (8.5 + 0.5) + 2s Using Associative Property of Addition
8.5 + (2s + 0.5)
= (8.5 + 0.5) + 2s Using Associative Property of Addition
= 9 + 2s Add
= 2(4.5) + 2(s) Using Distributive Property
= 2(4.5 + s) Using Associative Property of Addition
8.5 + (2s + 0.5) is equivalent to the following:
→ (8.5 + 2s) + 0.5
→ (8.5 + 0.5) + 2s
→ 2(4.5 + s)
Result
(8.5 + 2s) + 0.5
(8.5 + 0.5) + 2s
2(4.5 + s)
Page 40 Exercise 25 Answer
5n + 20
= 5(n) + 5(4) Using Distributive Property
= 5(n + 4)
15 + 5n + 5
= 20 + 5n Add
= 5(4) + 5(n) Using Distributive Property
= 5(n + 4) 5 is the common factor
5(n + 3) + 5
= 5(n) + 5(3) + 5 Using Distributive Property
= 5n + 15 + 5 Multiply
= 5n + 20 Add
= 5(n) + 5(4) Using Distributive Property
= 5(n + 4) 5 is the common factor
Result
5n + 20 is equivalent to the following:
→ 5n + 20
→ 15 + 5n + 5
→ 5(n + 3) + 5