Certainly! Let's go through each step of the solution process and justify each step with the appropriate property:
\begin{tabular}{|c|c|}
\hline
[tex]$2x + 5 = -7(x - 2)$[/tex] & original equation \\
\hline
[tex]$2x + 5 = -7x + 14$[/tex] & distributive property \\
\hline
[tex]$5 = -9x + 14$[/tex] & subtraction property of equality \\
\hline
[tex]$-9 = -9x$[/tex] & subtraction property of equality \\
\hline
[tex]$1 = x$[/tex] & division property of equality \\
\hline
[tex]$x = 1$[/tex] & symmetry property \\
\hline
\end{tabular}
### Explanation of Each Step:
1. Original Equation: The given equation is [tex]\(2x + 5 = -7(x - 2)\)[/tex].
2. Distributive Property: When we distribute the [tex]\(-7\)[/tex] across the parentheses on the right side, we apply the distributive property:
[tex]\[
-7(x - 2) = -7x + 14
\][/tex]
Thus, the equation becomes:
[tex]\[
2x + 5 = -7x + 14
\][/tex]
3. Subtraction Property of Equality: To isolate terms involving [tex]\(x\)[/tex], we add [tex]\(7x\)[/tex] to both sides:
[tex]\[
2x + 7x + 5 = 14
\][/tex]
Simplifying, we get:
[tex]\[
9x + 5 = 14
\][/tex]
Then we subtract 5 from both sides to isolate [tex]\(9x\)[/tex]:
[tex]\[
9x + 5 - 5 = 14 - 5
\][/tex]
Simplifying, we get:
[tex]\[
9x = 9
\][/tex]
4. Subtraction Property of Equality: Continuing from the previous step:
[tex]\[
9x + 5 = 14
\][/tex]
Simplifying further by subtracting 14 from both sides:
[tex]\[
5 - 14 = -9x
\][/tex]
Simplifying:
[tex]\[
-9 = -9x
\][/tex]
5. Division Property of Equality: To isolate [tex]\(x\)[/tex], we divide both sides by [tex]\(-9\)[/tex]:
[tex]\[
\frac{-9}{-9} = x
\][/tex]
Simplifying:
[tex]\[
1 = x
\][/tex]
6. Symmetry Property: Rewriting the final solution with [tex]\(x\)[/tex] on the left side gives:
[tex]\[
x = 1
\][/tex]
And that's the detailed, step-by-step solution to the equation!
