Complete the table by filling in the missing algebraic steps...

Complete the table by filling in the missing algebraic steps to solve the equation or supply the missing reasons for those steps. $$ 6 x+1=-2(2 x+1)-3 $$ $$ \begin{array}{|c|l|} \hline \begin{array}{c} \text { Algebraic Step to Solve } \\ \text { the Equation } \end{array} & {\text { Reason for Each Step }} \\ \hline & \begin{array}{l} \text { Simplify the right side } \\ \text { by using the distributive } \\ \text { property. } \end{array} \\ \hline \begin{array}{l} 6 x+1=-4 x-2-3 \\ 6 x+1=-4 x-5 \end{array} & \begin{array}{l} \end{array} \\ \hline & \begin{array}{l} \text { Move the variable terms to } \\ \text { one side using the addition } \\ \text { property of equality. } \end{array} \\ \hline 10 x+1=-5 & \\ -1-1 & \\ \hline & \begin{array}{l} \text { Solve for the variable } \\ \text { using the division property } \\ \text { of equality. } \end{array} \\ \hline x=-\frac{6}{10}=-\frac{3}{5} & \\ \hline \end{array} $$

Complete the table by filling in the missing algebraic steps to solve the equation or supply the missing reasons for those steps.
$$
6 x+1=-2(2 x+1)-3
$$
$$
\begin{array}{|c|l|}
\hline \begin{array}{c}
\text { Algebraic Step to Solve } \\
\text { the Equation }
\end{array} & {\text { Reason for Each Step }} \\
\hline & \begin{array}{l}
\text { Simplify the right side } \\
\text { by using the distributive } \\
\text { property. }
\end{array} \\
\hline \begin{array}{l}
6 x+1=-4 x-2-3 \\
6 x+1=-4 x-5
\end{array} & \begin{array}{l}
\end{array} \\
\hline & \begin{array}{l} \text { Move the variable terms to } \\
\text { one side using the addition } \\
\text { property of equality. } \end{array} \\
\hline 10 x+1=-5 & \\
-1-1 & \\
\hline & \begin{array}{l}
\text { Solve for the variable } \\
\text { using the division property } \\
\text { of equality. }
\end{array} \\
\hline x=-\frac{6}{10}=-\frac{3}{5} & \\
\hline
\end{array}
$$

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Key Concepts

Distributive Property

This property allows you to multiply a single term by each term inside a parenthesis, thereby eliminating the parentheses. It is essential for rewriting expressions in an equivalent form that can be more easily simplified or solved.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variables raised to the same power. This process simplifies expressions and is critical in reducing equations to a form where the variable can be easily isolated.

Addition Property of Equality

This property states that you can add the same value to both sides of an equation without changing the equality. It is used to move terms from one side of the equation to the other, helping to isolate variable terms.

Division Property of Equality

This property allows you to divide both sides of an equation by the same non-zero number, which is instrumental in isolating the variable by removing its coefficient.

Solving Linear Equations

This is a systematic process that involves applying algebraic operations such as distribution, combining like terms, and using the properties of equality to isolate and solve for the variable. It is a foundational skill in algebra that ensures any linear equation can be transformed into a simpler form to find the solution.

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