Answer :
Sure, let's carefully examine each of the steps and identify the missing justifications.
Given the equation [tex]\(2x - 7 = 15\)[/tex], we aim to solve for [tex]\(x\)[/tex]. Here's a step-by-step breakdown with detailed justifications:
1. Given: [tex]\(2x - 7 = 15\)[/tex]
This is the original equation provided.
2. Addition Property of Equality: [tex]\(2x - 7 + 7 = 15 + 7\)[/tex]
To isolate the term with [tex]\(x\)[/tex], we add 7 to both sides of the equation. The Addition Property of Equality states that if you add the same number to both sides of an equation, the sides remain equal.
3. Simplification: [tex]\(2x + 0 = 22\)[/tex]
After adding 7 to both sides, the left-hand side simplifies to [tex]\(2x + 0\)[/tex], and the right-hand side simplifies to 22.
(a) Justification (a): Additive Identity
The Additive Identity property states that adding zero to any number doesn’t change the value of the number. Therefore, [tex]\(2x + 0 = 2x\)[/tex].
4. Simplification: [tex]\(2x = 22\)[/tex]
Using the Additive Identity property simplifies [tex]\(2x + 0\)[/tex] to just [tex]\(2x\)[/tex].
5. Multiplication Property of Equality: [tex]\(\left(\frac{1}{2}\right)2x = \left(\frac{1}{2}\right)22\)[/tex]
Next, we multiply both sides by [tex]\(\frac{1}{2}\)[/tex] to isolate [tex]\(x\)[/tex]. The Multiplication Property of Equality states that if you multiply both sides of an equation by the same nonzero number, the sides remain equal.
(b) Justification (b): Division Property of Equality
Alternatively, multiplying both sides by [tex]\(\frac{1}{2}\)[/tex] is the same as dividing both sides by 2. This employs the Division Property of Equality, which states that dividing both sides of an equation by the same nonzero number does not change the equality.
6. Simplification: [tex]\(1x = 11\)[/tex]
After multiplying by [tex]\(\frac{1}{2}\)[/tex], the left-hand side becomes [tex]\(1x\)[/tex] and the right-hand side becomes 11.
This utilizes the Multiplicative Inverse property, as [tex]\(\frac{1}{2} \cdot 2 = 1\)[/tex].
7. Simplification: [tex]\(x = 11\)[/tex]
Finally, [tex]\(1x\)[/tex] is simply [tex]\(x\)[/tex] which gives us the solution [tex]\(x = 11\)[/tex].
This step uses the Multiplicative Identity, which states that any number multiplied by 1 is itself.
Putting all justifications together, we have:
[tex]\[ \begin{tabular}{|c|c|} \hline Step & Justification \\ \hline $2 x-7=15$ & Given \\ \hline $2 x-7+7=15+7$ & Addition Property of Equality \\ \hline $2 x+0=22$ & (a) Additive Identity \\ \hline $2 x=22$ & Additive Identity \\ \hline $\left(\frac{1}{2}\right) 2 x=\left(\frac{1}{2}\right) 22$ & (b) Division Property of Equality \\ \hline $1 x=11$ & Multiplicative Inverse \\ \hline $x=11$ & Multiplicative Identity \\ \hline \end{tabular} \][/tex]
Thus, the missing justifications are:
(a) Additive Identity
(b) Division Property of Equality
Given the equation [tex]\(2x - 7 = 15\)[/tex], we aim to solve for [tex]\(x\)[/tex]. Here's a step-by-step breakdown with detailed justifications:
1. Given: [tex]\(2x - 7 = 15\)[/tex]
This is the original equation provided.
2. Addition Property of Equality: [tex]\(2x - 7 + 7 = 15 + 7\)[/tex]
To isolate the term with [tex]\(x\)[/tex], we add 7 to both sides of the equation. The Addition Property of Equality states that if you add the same number to both sides of an equation, the sides remain equal.
3. Simplification: [tex]\(2x + 0 = 22\)[/tex]
After adding 7 to both sides, the left-hand side simplifies to [tex]\(2x + 0\)[/tex], and the right-hand side simplifies to 22.
(a) Justification (a): Additive Identity
The Additive Identity property states that adding zero to any number doesn’t change the value of the number. Therefore, [tex]\(2x + 0 = 2x\)[/tex].
4. Simplification: [tex]\(2x = 22\)[/tex]
Using the Additive Identity property simplifies [tex]\(2x + 0\)[/tex] to just [tex]\(2x\)[/tex].
5. Multiplication Property of Equality: [tex]\(\left(\frac{1}{2}\right)2x = \left(\frac{1}{2}\right)22\)[/tex]
Next, we multiply both sides by [tex]\(\frac{1}{2}\)[/tex] to isolate [tex]\(x\)[/tex]. The Multiplication Property of Equality states that if you multiply both sides of an equation by the same nonzero number, the sides remain equal.
(b) Justification (b): Division Property of Equality
Alternatively, multiplying both sides by [tex]\(\frac{1}{2}\)[/tex] is the same as dividing both sides by 2. This employs the Division Property of Equality, which states that dividing both sides of an equation by the same nonzero number does not change the equality.
6. Simplification: [tex]\(1x = 11\)[/tex]
After multiplying by [tex]\(\frac{1}{2}\)[/tex], the left-hand side becomes [tex]\(1x\)[/tex] and the right-hand side becomes 11.
This utilizes the Multiplicative Inverse property, as [tex]\(\frac{1}{2} \cdot 2 = 1\)[/tex].
7. Simplification: [tex]\(x = 11\)[/tex]
Finally, [tex]\(1x\)[/tex] is simply [tex]\(x\)[/tex] which gives us the solution [tex]\(x = 11\)[/tex].
This step uses the Multiplicative Identity, which states that any number multiplied by 1 is itself.
Putting all justifications together, we have:
[tex]\[ \begin{tabular}{|c|c|} \hline Step & Justification \\ \hline $2 x-7=15$ & Given \\ \hline $2 x-7+7=15+7$ & Addition Property of Equality \\ \hline $2 x+0=22$ & (a) Additive Identity \\ \hline $2 x=22$ & Additive Identity \\ \hline $\left(\frac{1}{2}\right) 2 x=\left(\frac{1}{2}\right) 22$ & (b) Division Property of Equality \\ \hline $1 x=11$ & Multiplicative Inverse \\ \hline $x=11$ & Multiplicative Identity \\ \hline \end{tabular} \][/tex]
Thus, the missing justifications are:
(a) Additive Identity
(b) Division Property of Equality
