Question

prove 4t + t = 5t
4t + t
t(4 + 1)
t(5)
5t
what property or properties were used to prove that the expressions are equivalent?
○ commutative and distributive properties
○ associative and distributive properties
○ associative and commutative properties
○ associative property

Explanation:

Step1: Analyze the proof of \(4t + t = 5t\)

First, rewrite \(t\) as \(1\times t\) (implicitly, since \(t = 1\times t\)). Then, using the distributive property of multiplication over addition, which states that \(a\times c + b\times c=(a + b)\times c\). Here, \(a = 4\), \(b = 1\), and \(c = t\), so \(4t + 1\times t=(4 + 1)t\).

Step2: Simplify \((4 + 1)t\)

Calculate \(4+1 = 5\), so \((4 + 1)t = 5t\). Also, the step of writing \(t\) as \(1\times t\) and then combining like terms can be related to the commutative property (though more prominently, the distributive property is key here, and also, the commutative property of multiplication is implicit in the fact that \(1\times t=t\times1\), but the main property used for combining \(4t\) and \(t\) is the distributive property, and also, the commutative property in the sense of re - ordering the terms (since \(t+4t = 4t + t\) by commutative property of addition) and then distributive. Wait, let's re - examine: \(4t+t=4t + 1t\) (by identity property of multiplication, \(t = 1\times t\)), then by distributive property \(a\times c + b\times c=(a + b)\times c\), here \(a = 4\), \(b = 1\), \(c=t\), so \((4 + 1)t=5t\). Also, the commutative property of addition allows us to write \(t + 4t=4t + t\), but the key properties to get from \(4t + t\) to \(5t\) are commutative (for addition, to ensure the order doesn't matter) and distributive (to factor out \(t\)). So the properties used are commutative and distributive.

Answer:

A. commutative and distributive properties