Question

determining equivalent expressions
which expressions are equivalent to the expression 7(5 + m) − 8? check all that apply.
□ 7m + 27
□ 34m
□ 35 + m − 8
□ 35 + 7m − 8
□ 75 + m − 8

Explanation:

Step1: Simplify the original expression

First, expand \(7(5 + m)-8\) using the distributive property \(a(b + c)=ab+ac\). So \(7\times5+7\times m - 8=35 + 7m-8\).

Step2: Further simplify the expanded expression

Combine the constant terms: \(35-8 = 27\), so \(35 + 7m-8=7m + 27\). Also, \(35 + 7m-8\) can be rewritten as \(35+m - 8+6m\)? Wait, no, wait. Wait, the original expansion is \(35 + 7m-8\), and also, let's check \(75+m - 8\): no, wait, \(35-8 = 27\), \(7\times5 = 35\), and if we consider another way: Wait, \(7(5 + m)-8=35+7m - 8\), and also, let's compute \(35 - 8=27\), so \(7m + 27\) is equivalent. Also, \(35+7m - 8\) is the expanded form. Wait, and what about \(75+m - 8\)? No, that's not. Wait, let's re - check:

Original expression: \(7(5 + m)-8\)

Expand: \(7\times5+7\times m-8=35 + 7m-8\) (this is one equivalent expression)

Then, combine like terms in \(35 + 7m-8\): \(35-8 = 27\), so \(7m+27\) (this is another equivalent expression)

Now check the options:

• Option 1: \(7m + 27\): Equivalent (as shown above)
• Option 2: \(34m\): \(7(5 + m)-8=35 + 7m-8=7m + 27

eq34m\) (unless \(m\) has a specific value, but generally not equivalent)

• Option 3: \(35 + m-8\): \(35 + m-8=27 + m\), while \(7(5 + m)-8=35 + 7m-8=7m + 27\), not equivalent (since the coefficient of \(m\) is different)
• Option 4: \(35 + 7m-8\): Equivalent (this is the expanded form)
• Option 5: \(75+m - 8\): \(75-8=67\), so \(m + 67\), which is not equivalent to \(7m+27\)

Wait, I made a mistake earlier. Let's recalculate:

\(7(5 + m)-8=35 + 7m-8\) (by distributive property)

Then, \(35-8 = 27\), so \(35 + 7m-8=7m + 27\) (combining constants)

Also, \(35 + 7m-8\) is the same as the expanded form, so the equivalent expressions are \(7m + 27\), \(35 + 7m-8\)

Answer:

The equivalent expressions are:

• \(7m + 27\)
• \(35 + 7m-8\)