Question

Ethan's solution and reasoning for solving an equation are shown below: Given: Four halves times x minus ten equals thirty Step 1: Four halves times x minus ten equals thirty Reason 1: Given Step 2: Four halves times x minus ten plus ten equals thirty plus ten Reason 2: Addition Property of Equality Step 3: Four halves times x equals forty Reason 3: Simplify Step 4: Four halves times x times two fourths equals forty times two fourths Reason 4: Division Property of Equality Step 5: x = 20 Reason 5: Simplify Choose which reason is incorrect. Reason 1 Reason 2 Reason 3 Reason 4

Answer 1

Reason 4

Explanation:

We are given that

`(4)/(2)x-10=30`

Step 1:
`(4)/(2)x-10=30`

Reason: Given

Step 2:
`(4)/(2)x-10+10=30+10`

Reason :Addition property of equality

Step 3:
`(4)/(2)x=40`

Reason: Simplify

Step 4:
`(4)/(2)x}* (2)/(4)=40* (2)/(4)`

Reason: Multiply by 2/4 on both sides

Step 5:
`x=20`

Reason: Simplify

Hence, reason 4 is incorrect because in step 4 he used division property of equality instead of mutiplication by 2/4.

Answer 2

Step 4 - Reason 4: INVALID

Explanation:

This problem is a typical case of solving an algebraic expression of one uknown variable, which here is denoted with
`x`.

Let us first rewrite all information denoted in the question as follow:

Given:
`(4)/(2)x-10=30`

Step 1:
`(4)/(2)x-10=30` Reason 1: Given

Step 2:
`(4)/(2)x-10+10=30+10` Reason 2: Addition Property of Equality

Step 3:
`(4)/(2)x=40` Reason 3: Simplify

Step 4:
`(4)/(2)x*(2)/(4)=40*(2)/(4)` Reason 4: Division Property of Equality

Step 5:
`x=20` Reason 5: Simplify

Having obtained all solution steps we can conclude the following:

Step 1 - Reason 1: Valid because this is our given equation to be solved

Step 2 - Reason 2: Valid because this is a typical first step in such equations to eliminate our constants from the side (here the Left Hand Side) and have only our Uknown variables (in this case
`x`).

Step 3 - Reason 3: Valid because we just compute the addition/subtractions on each side to simplify the expression

Step 4 - Reason 4: INVALID because at this point we already have one Unknown variable on the left and one constant on the right. So all we have to do is simplify the expression as it is. So here instead we should actually have:

`(4)/(2)x=40` ⇔
`4x=2*40` ⇔
`4x=80`⇔
`x=(80)/(4)`⇔
`x=20`

OR we could also solve it as:

`(4)/(2)x=40` ⇔
`2x=40` ⇔
`x=(40)/(2)` ⇔
`x=20`

Which essentially is Step 5 - Reason 5: Valid because of the computations just above.