Question

Study the steps used to solve the equation. Given: StartFraction c Over 2 EndFraction minus 5 equals 7 Step 1: StartFraction c Over 2 EndFraction minus 5 plus 5 equals 7 plus 5 Step 2: StartFraction c Over 2 EndFraction plus 0 equals 12 Step 3: StartFraction c Over 2 EndFraction equals 12 Step 4: 2 (StartFraction c Over 2 EndFraction) equals 12 (2) Step 5: c equals 24 Choose the property that justifies each step of the solution. Step 1: Step 2: Step 3: Step 4:

Answer 1

1. addition property of equality
2. integers are closed to addition
3. identity element
4. multiplication property of equality
5. commutative property of multiplication; reals are closed to multiplication; identity element

Explanation:

Given:

c/2 -5 = 7

Step 1: c/2 -5 +5 = 7 +5

Step 2: c/2 +0 = 12

Step 3: c/2 = 12

Step 4: 2(c/2) = 12(2)

Step 5: c = 24

Find:

The property that justifies each step of the solution.

Solution:

Step 1: addition property of equality (lets you add the same to both sides)

Step 2: integers are closed to addition

Step 3: identity property of addition (adding 0 changes nothing)

Step 4: multiplication property of equality

Step 5: closure of real numbers to multiplication; identity property of multiplication

_____

It is hard to say what "property" you want to claim when you simplify an arithmetic expression. Above, we have used the property that the sets of integers and real numbers are closed to addition and multiplication. That is, adding or multiplying real numbers gives a real number.

In Step 5, we can rearrange 2(c/2) to c(2/2) using the commutative property of multiplication. 2/2=1, and c×1 = c. The latter is due to the identity element for multiplication: multiplying by 1 changes nothing.

Apart from the arithmetic, the other properties used are properties of equality. Those let you perform any operation on an equation, as long as you do it to both sides of the equation. The operations we have performed in this fashion are adding 5 and multiplying by 2.

Answer 2

Step 1 ~ addition property of equality

Step 2 ~ additive inverses

Step 3 ~ additive identity

Step 4 ~ multiplication property of equality

Explanation:

Addition property of equality means that If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal. In this problem, they added 5 to both sides to make the equation balanced.

Additive inverses means what you add to a number to get zero. The negative of a number. -5 + 5 = 0.

Additive identity means that the sum of a number and 0 is that number.

Multiplication property of equality states that when you multiply both sides of an equation by the same number, the two sides remain equal. In this problem, they multiplied 2 to both sides to get rid of the denominator in the fraction.