Question

emily is solving a linear equation. her work is shown below:
step 1: (-2(x + 5) = 4x + 8)
step 2: (-2x - 10 = 4x + 8)
step 3: (-2x - 10 - 8 = 4x + 8 - 8)

from step 1 to step 2, what field axiom for the real numbers did emily use?
select

in step 3, emily subtracted 8 from both sides because it will equal (4x + 0) on the right. 0 is the select element for the field of real numbers.

Explanation:

First Sub - Question (From Step 1 to Step 2)

Step 1: Analyze the transformation

In Step 1, we have \(-2(x + 5)\), and in Step 2, it becomes \(-2x-10\). This is done by applying the distributive property (also known as the distributive axiom of real numbers), which states that \(a(b + c)=ab+ac\). Here, \(a=-2\), \(b = x\) and \(c = 5\), so \(-2(x + 5)=-2\times x+(-2)\times5=-2x - 10\).

Step 2: Identify the axiom

The transformation from \(-2(x + 5)\) to \(-2x-10\) uses the distributive property (distributive axiom) of real numbers.

Second Sub - Question (0 as an element)

Step 1: Recall the definition of additive identity

The additive identity element of a set (in this case, the set of real numbers) is an element \(e\) such that for any element \(a\) in the set, \(a+e=a\) and \(e + a=a\). For real numbers, when we add 0 to any real number \(a\) (i.e., \(a + 0=a\) and \(0 + a=a\)), the result is the number itself. In Step 3, when we subtract 8 from both sides, on the right - hand side \(4x+8 - 8=4x+(8 - 8)=4x + 0\), and \(4x+0 = 4x\) (because 0 is the additive identity).

Step 2: Identify the element type

0 is the additive identity element for the field of real numbers.

Answer:

First Sub - Question: Distributive Property (Distributive Axiom)
Second Sub - Question: Additive Identity