Question

which expressions are equivalent to $-20f - 16 + 24f - 36$? choose all that apply. $-4(-5f - 4 + 6f - 9)$ $-4(-f + 13)$ $-4(-11f - 5)$ $-4(5f - 4 + 6f - 9)$ $-4(5f + 4 - 6f + 9)$

Explanation:

Step1: Simplify the original expression

First, combine like terms in \(-20f - 16 + 24f - 36\). Combine the \(f\)-terms: \(-20f+24f = 4f\). Combine the constant terms: \(-16-36=-52\). So the original expression simplifies to \(4f - 52\). We can also factor out \(-4\) from the original expression: \(-20f - 16 + 24f - 36=-4(5f + 4 - 6f + 9)\) (let's check this later) or factor out \(4\) as \(4(f - 13)\) or factor out \(-4\) as \(-4(-f + 13)\) (since \(4(f - 13)=-4(-f + 13)\)).

Step2: Analyze each option

Option 1: \(-4(-5f - 4 + 6f - 9)\)

Simplify inside the parentheses: \(-5f+6f - 4 - 9=f - 13\). Then multiply by \(-4\): \(-4(f - 13)=-4f + 52\), which is not equal to \(4f - 52\).

Option 2: \(-4(-f + 13)\)

Distribute the \(-4\): \((-4)\times(-f)+(-4)\times13 = 4f-52\), which matches the simplified original expression.

Option 3: \(-4(-11f - 5)\)

Distribute the \(-4\): \((-4)\times(-11f)+(-4)\times(-5)=44f + 20\), which is not equal to \(4f - 52\).

Option 4: \(-4(5f - 4 + 6f - 9)\)

Simplify inside the parentheses: \(5f+6f - 4 - 9 = 11f-13\). Multiply by \(-4\): \(-44f + 52\), not equal to \(4f - 52\).

Option 5: \(-4(5f + 4 - 6f + 9)\)

Simplify inside the parentheses: \(5f-6f + 4 + 9=-f + 13\). Multiply by \(-4\): \(-4\times(-f)+(-4)\times13 = 4f-52\), which matches the simplified original expression.

Answer:

The equivalent expressions are:

• \(-4(-f + 13)\)
• \(-4(5f + 4 - 6f + 9)\)