Question

nancy wrote the expression 3x - 12 to represent the relationship in a table of values. use properties of operations to write two equivalent expressions.

Explanation:

Step1: Factor out the greatest common factor

The greatest common factor of \(3x\) and \(-12\) is \(3\). Using the distributive property \(ab + ac = a(b + c)\) (in reverse, factoring), we get \(3(x - 4)\) since \(3\times x=3x\) and \(3\times(- 4)=-12\).

Step2: Rearrange terms (or use other properties)

We can also rewrite the expression by splitting the coefficient or using the commutative property. For example, we can write it as \(3x+(- 12)\) (which is just the original expression rewritten to show the subtraction as addition of a negative), or we can factor out a different common factor (though \(3\) is the greatest, we could also factor out \(1\) or \(- 3\)). Let's use factoring out \(-3\): \(-3(-x + 4)\) (since \(-3\times(-x)=3x\) and \(-3\times4=-12\)). Another way: we can rewrite it as \(x+2x - 12\) (by splitting \(3x\) into \(x + 2x\)) or \(3x-10 - 2\) (splitting \(-12\) into \(-10-2\)).

Answer:

Two equivalent expressions are \(3(x - 4)\) and \(x+2x - 12\) (answers may vary, other valid equivalents include \(3x+(-12)\), \(-3(-x + 4)\), etc.)