Let's solve the given equation step-by-step to verify Lily's process:
1. Start with the given equation:
[tex]\[
3x - 10 = -3x + 8
\][/tex]
2. Use the addition property of equality to combine like terms involving [tex]\( x \)[/tex] on one side of the equation. Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
3x - 10 + 3x = -3x + 8 + 3x
\][/tex]
Simplifying, this becomes:
[tex]\[
6x - 10 = 8
\][/tex]
This corresponds to Lily's first step:
[tex]\[
\text{I used the addition property of equality to achieve } 6x - 10 = 8.
\][/tex]
3. Next, use the addition property of equality to isolate the term with [tex]\( x \)[/tex]. Add 10 to both sides:
[tex]\[
6x - 10 + 10 = 8 + 10
\][/tex]
Simplifying, this gives:
[tex]\[
6x = 18
\][/tex]
This corresponds to Lily's second step:
[tex]\[
\text{I used the addition property of equality to achieve } 6x = 18.
\][/tex]
4. Finally, use the division property of equality to solve for [tex]\( x \)[/tex]. Divide both sides by 6:
[tex]\[
\frac{6x}{6} = \frac{18}{6}
\][/tex]
Simplifying, this results in:
[tex]\[
x = 3
\][/tex]
Note that there is a slight inconsistency in Lily's justification here. Instead of saying "I used the subtraction property of equality," it would be more accurate to say:
[tex]\[
\text{I used the division property of equality to achieve the solution of } x = 3.
\][/tex]
Thus, to answer the question about the justification of steps, you can select:
1. I used the addition property of equality to achieve [tex]\( 6x - 10 = 8 \)[/tex].
2. I used the addition property of equality to achieve [tex]\( 6x = 18 \)[/tex].
