Question

are the expressions $5x + 3x$ and $8x$ equivalent? explain.

problem 36

use the distributive property to rewrite: $6(a + 4)$

problem 37

which expression is equivalent to $3(x + 5)$?
a) $3x + 5$
b) $3x + 15$
c) $x + 15$
d) $3x + 3$

problem 38

evaluate: $2 + 3 \times 4 - 2^2$

Explanation:

First Question: Are the expressions \(5x + 3x\) and \(8x\) equivalent? Explain.

Step1: Combine like terms in \(5x + 3x\)

Like terms have the same variable part. For \(5x\) and \(3x\), we add their coefficients. The coefficient of \(5x\) is \(5\) and of \(3x\) is \(3\). So \(5 + 3 = 8\), and when we add the variable part \(x\) back, we get \(8x\).

Step2: Compare the two expressions

The first expression \(5x + 3x\) simplifies to \(8x\) (from Step 1), and the second expression is \(8x\). Since simplifying \(5x + 3x\) gives \(8x\), the two expressions are equivalent.

Step1: Recall the distributive property

The distributive property states that \(c(a + b)=ca+cb\), where \(c\), \(a\), and \(b\) are real numbers (or algebraic expressions).

Step2: Apply the distributive property to \(6(a + 4)\)

Here, \(c = 6\), \(a=a\), and \(b = 4\). So we multiply \(6\) with \(a\) and \(6\) with \(4\) separately.
\(6\times a=6a\) and \(6\times4 = 24\). Then we add these two results together.
So \(6(a + 4)=6a+24\).

Step1: Apply the distributive property

The distributive property is \(c(a + b)=ca+cb\). For \(3(x + 5)\), \(c = 3\), \(a=x\), and \(b = 5\).

Step2: Multiply and simplify

Multiply \(3\) with \(x\) and \(3\) with \(5\). \(3\times x=3x\) and \(3\times5 = 15\). Then add these results: \(3x+15\).
Now we check the options:

• Option A: \(3x + 5\) (incorrect, since \(3\times5 = 15\) not \(5\))
• Option B: \(3x + 15\) (matches our result)
• Option C: \(x + 15\) (incorrect, the coefficient of \(x\) should be \(3\) not \(1\))
• Option D: \(3x + 3\) (incorrect, \(3\times5 = 15\) not \(3\))

Answer:

Yes, the expressions \(5x + 3x\) and \(8x\) are equivalent because combining the like terms in \(5x + 3x\) (adding the coefficients \(5\) and \(3\)) gives \(8x\), which is the same as the second expression.

Problem 36: Use the distributive property to rewrite: \(6(a + 4)\)