\begin{boxdef}
    \large{\textbf{Equation:}}
    \begin{solutionordottedlines}[1in]
    \end{solutionordottedlines}
\end{boxdef}

\begin{boxdef}
    \large{\textbf{Equality of equations:}}
    \begin{solutionordottedlines}[1in]
        What does it mean for 2 equations to be equal? It means that we can obtain the other equation by applying some combination of operations to the first equation. 
    \end{solutionordottedlines}
\end{boxdef}

\begin{demobox}
    Allow us to consider the following set of equations:
        \[
        \begin{aligned}
        & 2 x+7=11 \\
        & 2 x+9=13
        \end{aligned}
        \]

    Equation (2) is obtained from equation (1) by adding 2 to each side of the equation.

    So equation (1) is obtained from equation (2) by subtracting 2 from each side of the equation.

    Equations (1) and (2) are said to be equivalent equations.

    Equation (3), below, is obtained from equation (1) by subtracting 7 from each side of the equation.

        \[
        \begin{aligned}
        2 x & =4 \\
        x & =2
        \end{aligned}
        \]

    Equation (4) is obtained from equation (3) by dividing each side of the equation by 2. You can obtain equation (3) from equation (4) by multiplying each side of the equation by 2 .

    All of the above equations are equivalent.
\end{demobox}

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \questionpoints[4] Solve:
        \begin{parts}\begin{multicols}{4}
            \part \(3 x+5=20\)
            \begin{solutionordottedlines}[1in]
                \(3 x+5=20\)

                \(3 x=15 \quad\) (Subtract 5 from both sides.)

                \(x=5 \quad\) (Divide both sides by 3 .)
            \end{solutionordottedlines}
            \part \(5 x-7=18\)
            \begin{solutionordottedlines}[1in]
                \(5 x-7=18\)

                \(5 x=25 \quad\) (Add 7 to both sides.)

                \(x=5 \quad\) (Divide both sides by 5.)
            \end{solutionordottedlines}
            \part \(3-2 x=15\)
            \begin{solutionordottedlines}[1in]
                \(3-2 x=15\)

                \(-2 x=12 \quad\) (Subtract 3 from both sides.)

                \(x=-6 \quad\) (Divide both sides by -2 .)
            \end{solutionordottedlines}
            \part \(-3 p=\frac{2}{5}\)
            \begin{solutionordottedlines}[1in]
                \(-3 p=\frac{2}{5}\)

                \[
                p=-\frac{2}{15} \quad \text { (Divide both sides by }-3 \text {.) }
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \questionpoints[2] Solve:
        \begin{parts}\begin{multicols}{2}
            \part \(3 x+7=2 x+13\) 
            \begin{solutionordottedlines}[1in]
                \[
                3 x+7=2 x+13
                \]

                \[
                \begin{aligned}
                3 x+7-2 x & =2 x+13-2 x & & \text { (Subtract } 2 x \text { from both sides.) } \\
                x+7 & =13 & & \\
                x & =6 & & \text { (Subtract } 7 \text { from both sides.) }
                \end{aligned}
                \]
            \end{solutionordottedlines}
            \part \(5 a-21=14-2 a\)
            \begin{solutionordottedlines}[1in]
                \(5 a-21=14-2 a\)

                \[
                \begin{aligned}
                7 a-21 & =14 & & \text { (Add } 2 a \text { to both sides.) } \\
                7 a & =35 & & \text { (Add } 21 \text { to both sides.) } \\
                a & =5 & & \text { (Divide both sides by 7.) }
                \end{aligned}
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
    \end{questions}
\end{examplebox}

\begin{exercisebox}
    \begin{questions}
        \questionpoints[14] Solve these equations.
        \begin{parts}\begin{multicols}{2}
            \part \(a+2=5\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(d-15=3\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(9 q=-27\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(x-6=5 \frac{1}{3}\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(2 a+5=7\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(5 d-7=23\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(5 h+21=11\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(3 a-16=-31\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(2 x+11=7 \frac{1}{4}\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(2-3 a=8\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(3-5 d=-22\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(5 x+5=3 x+1\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(5 x-6=x-2\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \part \(4 x-3=18-3 x\)
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
    \end{questions}
\end{exercisebox}
