To solve the given equation [tex]\(\frac{c}{2} - 5 = 7\)[/tex], let's identify the properties used at each step to justify the operations performed. Here is the detailed step-by-step solution:
Given:
[tex]\[ \frac{c}{2} - 5 = 7 \][/tex]
Step 1:
[tex]\[ \frac{c}{2} - 5 + 5 = 7 + 5 \][/tex]
In this step, we add 5 to both sides of the equation to isolate the term with the variable [tex]\(c\)[/tex]. This step uses the Addition Property of Equality, which states that if you add the same number to both sides of an equation, the two sides remain equal.
[tex]\[ \text{Step 1: } \boxed{\text{Addition Property of Equality}} \][/tex]
Step 2:
[tex]\[ \frac{c}{2} + 0 = 12 \][/tex]
In this step, we combine [tex]\(-5\)[/tex] and [tex]\(+5\)[/tex] on the left side, which simplifies to 0, thus:
[tex]\[ \frac{c}{2} = 12 \][/tex]
The simplification [tex]\(\frac{c}{2} + 0 = \frac{c}{2}\)[/tex] is straightforward and uses the Identity Property, which states that adding 0 to any number doesn't change its value.
[tex]\[ \text{Step 2: } \boxed{\text{Identity Property}} \][/tex]
Step 3:
[tex]\[ \frac{c}{2} = 12 \][/tex]
This step indicates that the equation is already simplified, confirming the previous result. Thus, this step also falls under the Identity Property, as no further operations are performed.
[tex]\[ \text{Step 3: } \boxed{\text{Identity Property}} \][/tex]
Step 4:
[tex]\[ 2 \left( \frac{c}{2} \right) = 12 \cdot 2 \][/tex]
In this step, we multiply both sides of the equation by 2 to solve for [tex]\(c\)[/tex]. This step uses the Multiplication Property of Equality, which states that multiplying both sides of an equation by the same nonzero number keeps the sides equal.
[tex]\[ \text{Step 4: } \boxed{\text{Multiplication Property of Equality}} \][/tex]
Step 5:
[tex]\[ c = 24 \][/tex]
Finally, we simplify the expression [tex]\(2 \left( \frac{c}{2} \right) = c\)[/tex] on the left side and calculate the right side to find [tex]\(c = 24\)[/tex].
So, the properties that justify each step are:
- Step 1: [tex]\(\boxed{\text{Addition Property of Equality}}\)[/tex]
- Step 2: [tex]\(\boxed{\text{Identity Property}}\)[/tex]
- Step 3: [tex]\(\boxed{\text{Identity Property}}\)[/tex]
- Step 4: [tex]\(\boxed{\text{Multiplication Property of Equality}}\)[/tex]
