Question

name: date: per: homework this is a 2 - page document! directions: write the letter of the property of equality that justifies each statement. 1. if a = 2b, then a - c = 2b - c 2. 8x = 8x 3. 3(p - 7) = 3p - 21 4. if -7k = -42, then k = 6 5. if m + n = 15 and n = 2, then m + 2 = 15 6. if \\(\frac{x}{4} = -5\\), x = -20 7. if \\(w^2 = 2x\\) and 2x = y, then \\(w^2 = y\\) 8. if c - 9 = -1, then c = 8 9. if n = -3, then -3 = n a. addition property of equality b. subtraction property of equality c. multiplication property of equality d. division property of equality e. distributive property f. substitution property g. reflexive property h. symmetric property i. transitive property ctions: complete each proof using the properties of equality. not all rows may be used. given: -8(x - 3) = -32; prove: x = 7 statements reasons

Explanation:

Part 1: Answering the Multiple - Choice Questions (1 - 9)

Question 1

• Brief Explanations: The Subtraction Property of Equality states that if \(a = b\), then \(a - c=b - c\). Here, we start with \(a = 2b\) and subtract \(c\) from both sides to get \(a - c=2b - c\), so the property is the Subtraction Property of Equality (B).
• Answer: B

Question 2

• Brief Explanations: The Reflexive Property of Equality states that for any real number \(x\), \(x=x\). Here, \(8x = 8x\) follows the Reflexive Property (G).
• Answer: G

Question 3

• Brief Explanations: The Distributive Property is \(a(b - c)=ab - ac\). Here, \(3(p - 7)=3\times p-3\times7 = 3p - 21\) which uses the Distributive Property (E).
• Answer: E

Question 4

• Brief Explanations: The Division Property of Equality states that if \(a = b\) and \(c

eq0\), then \(\frac{a}{c}=\frac{b}{c}\). Given \(- 7k=-42\), we divide both sides by \(-7\) (\(\frac{-7k}{-7}=\frac{-42}{-7}\)) to get \(k = 6\), so it's the Division Property of Equality (D).

• Answer: D

Question 5

• Brief Explanations: The Substitution Property of Equality allows us to substitute a value equal to a variable into an equation. Since \(n = 2\), we substitute \(2\) for \(n\) in \(m + n=15\) to get \(m + 2=15\) (F).
• Answer: F

Question 6

• Brief Explanations: The Multiplication Property of Equality states that if \(a = b\), then \(a\times c=b\times c\). Given \(\frac{x}{4}=-5\), we multiply both sides by \(4\) (\(\frac{x}{4}\times4=-5\times4\)) to get \(x=-20\), so it's the Multiplication Property of Equality (C).
• Answer: C

Question 7

• Brief Explanations: The Transitive Property of Equality states that if \(a = b\) and \(b = c\), then \(a = c\). Here, \(w^{2}=2x\) and \(2x = y\), so \(w^{2}=y\) (I).
• Answer: I

Question 8

• Brief Explanations: The Addition Property of Equality states that if \(a = b\), then \(a + c=b + c\). Given \(c-9=-1\), we add \(9\) to both sides (\(c-9 + 9=-1 + 9\)) to get \(c = 8\), so it's the Addition Property of Equality (A).
• Answer: A

Question 9

• Brief Explanations: The Symmetric Property of Equality states that if \(a = b\), then \(b = a\). Given \(n=-3\), then \(-3 = n\) (H).
• Answer: H

Part 2: Proving \(x = 7\) given \(-8(x - 3)=-32\)

Step 1: Apply the Distributive Property

We start with the given equation \(-8(x - 3)=-32\). Using the Distributive Property \(a(b - c)=ab - ac\) (where \(a=-8\), \(b = x\), \(c = 3\)), we get \(-8x+24=-32\).

Step 2: Apply the Subtraction Property of Equality

Subtract \(24\) from both sides of the equation \(-8x + 24=-32\). By the Subtraction Property of Equality (if \(a = b\), then \(a - c=b - c\)), we have \(-8x+24 - 24=-32 - 24\), which simplifies to \(-8x=-56\).

Step 3: Apply the Division Property of Equality

Divide both sides of the equation \(-8x=-56\) by \(-8\). By the Division Property of Equality (if \(a = b\) and \(c
eq0\), then \(\frac{a}{c}=\frac{b}{c}\)), we get \(\frac{-8x}{-8}=\frac{-56}{-8}\), so \(x = 7\).

Statements	Reasons
\(-8x + 24=-32\)	Distributive Property (E)
\(-8x+24 - 24=-32 - 24\)	Subtraction Property of Equality (B)
\(-8x=-56\)	Simplification
\(\frac{-8x}{-8}=\frac{-56}{-8}\)	Division Property of Equality (D)
\(x = 7\)	Simplification

Answer:

The proof shows that \(x = 7\) using the properties of equality as above.