To solve the given equation step by step and justify each step with the appropriate property, we start with the original equation and proceed as follows:
1. Original Equation:
[tex]\[
2x + 5 = -7(x - 2)
\][/tex]
This is the equation we need to solve for [tex]\( x \)[/tex].
2. Apply the Distributive Property:
[tex]\[
2x + 5 = -7x + 14
\][/tex]
Explanation: We use the distributive property to expand the right-hand side of the equation.
3. Combine Like Terms:
[tex]\[
2x + 5 = -7x + 14 \implies 5 = -9x + 14
\][/tex]
Explanation: To isolate the [tex]\( x \)[/tex] terms, we subtract [tex]\( 2x \)[/tex] from both sides of the equation. This simplifies the equation by combining terms involving [tex]\( x \)[/tex] on one side.
4. Apply the Subtraction Property of Equality:
[tex]\[
5 = -9x + 14 \implies -9 = -9x
\][/tex]
Explanation: We subtract 14 from both sides of the equation to isolate the term involving [tex]\( x \)[/tex]. This is known as the subtraction property of equality.
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
-9 = -9x \implies 1 = x
\][/tex]
Explanation: We divide both sides of the equation by -9 to solve for [tex]\( x \)[/tex].
6. Final Solution:
[tex]\[
x = 1
\][/tex]
Explanation: Rearranging [tex]\( 1 = x \)[/tex] to the more conventional form [tex]\( x = 1 \)[/tex].
Therefore, the step-by-step process justifying each step with the appropriate property is as demonstrated in the table:
\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] & combine like terms \\
\hline
[tex]$-9 = -9x$[/tex] & subtraction property of equality \\
\hline
[tex]$1 = x$[/tex] & division property of equality \\
\hline
[tex]$x = 1$[/tex] & simplify and rearrange \\
\hline
\end{tabular}
