Let's go through the steps used to solve the equation [tex]\(\frac{c}{2} - 5 = 7\)[/tex] and justify each step with the appropriate mathematical property.
Given equation:
[tex]\[
\frac{c}{2} - 5 = 7
\][/tex]
Step 1:
[tex]\[
\frac{c}{2} - 5 + 5 = 7 + 5
\][/tex]
Justification for Step 1:
In this step, we are adding 5 to both sides of the equation to isolate the term with [tex]\(c\)[/tex]. The property used here is the addition property of equality, which states that if you add the same amount to both sides of an equation, the two sides remain equal.
[tex]\[
\text{Step 1 Justification: } \boxed{\text{addition property of equality}}
\][/tex]
Step 2:
[tex]\[
\frac{c}{2} + 0 = 12
\][/tex]
Justification for Step 2:
In this step, adding -5 and 5 gives us 0, because they are additive inverses. The property used here is additive inverses, which states that a number plus its inverse equals zero.
[tex]\[
\text{Step 2 Justification: } \boxed{\text{additive inverses}}
\][/tex]
Step 3:
[tex]\[
\frac{c}{2} = 12
\][/tex]
Justification for Step 3:
Adding 0 to a number does not change its value. This step uses the additive identity property, which states that any number plus zero is the number itself.
[tex]\[
\text{Step 3 Justification: } \boxed{\text{additive identity}}
\][/tex]
Step 4:
[tex]\[
2\left(\frac{c}{2}\right) = 12 \times 2
\][/tex]
Justification for Step 4:
In this step, we are multiplying both sides of the equation by 2 to isolate [tex]\(c\)[/tex]. The property used here is the multiplication property of equality, which states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal.
[tex]\[
\text{Step 4 Justification: } \boxed{\text{multiplication property of equality}}
\][/tex]
Step 5:
[tex]\[
c = 24
\][/tex]
Here, we find the value of [tex]\(c\)[/tex] to be 24 after performing the multiplication on both sides.
Summarizing the properties used for each step:
1. Step 1: [tex]\(\boxed{\text{addition property of equality}}\)[/tex]
2. Step 2: [tex]\(\boxed{\text{additive inverses}}\)[/tex]
3. Step 3: [tex]\(\boxed{\text{additive identity}}\)[/tex]
4. Step 4: [tex]\(\boxed{\text{multiplication property of equality}}\)[/tex]
