Question

1. which of the following is an equivalent form of the expression below? ( 8r - 5(2r + 3) + 6r - 12 ) circle all that apply. (a) ( 8r - 10r - 15 + 6r - 12 ) (b) ( -4r - 27 ) (c) ( 4r + 3 ) (d) ( 4r - 27 )

Explanation:

First, we simplify the original expression \(8r - 5(2r + 3)+6r - 12\).

Step 1: Distribute the -5

We use the distributive property \(a(b + c)=ab+ac\). Here, \(a = - 5\), \(b=2r\) and \(c = 3\). So we have:
\(8r-5\times2r-5\times3 + 6r-12=8r-10r - 15+6r-12\)

Step 2: Combine like terms (r - terms)

Combine the terms with \(r\): \(8r-10r + 6r=(8 - 10+6)r=4r\)

Step 3: Combine like terms (constant terms)

Combine the constant terms: \(-15-12=-27\)

Step 4: Combine the results

After combining like terms, we get \(4r-27\)

Now let's check each option:

Option (a)

The expression is \(8r-10r - 15+6r-12\), which is the intermediate step after distribution. Let's simplify it:
Combine \(r\) terms: \(8r-10r + 6r = 4r\)
Combine constants: \(-15-12=-27\)
So \(8r-10r - 15+6r-12=4r - 27\), so (a) is equivalent.

Option (b)

The expression is \(- 4r-27\). From our simplification, we have \(4r-27\), which is not equal to \(-4r - 27\) (unless \(r = 0\), but in general, they are not equivalent). So (b) is not equivalent.

Option (c)

The expression is \(4r + 3\). Our simplified expression is \(4r-27\), which is not equal to \(4r + 3\) (the constant terms are different). So (c) is not equivalent.

Option (d)

The expression is \(4r-27\), which is exactly the simplified form of the original expression. So (d) is equivalent.

Answer:

a. \(8r - 10r - 15 + 6r - 12\), d. \(4r - 27\)