Activity 1.1.1

With a small number of unknowns, we are able to graph the sets of solutions to linear equations. Here, we will consider collections of equations having two unknowns.

  1. On the plot below, graph the lines

    \begin{equation*} \begin{aligned} y \amp = x+1 \\ y \amp = 2x-1\text{.} \\ \end{aligned} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy both equations?

  2. On the plot below, graph the lines

    \begin{equation*} \begin{aligned} y \amp = x+1 \\ y \amp = x-1\text{.} \\ \end{aligned} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy both equations?

  3. On the plot below, graph the line

    \begin{equation*} y = x+1\text{.} \end{equation*}

    How many points \((x,y)\) satisfy this equation?

  4. On the plot below, graph the lines

    \begin{equation*} \begin{aligned} y \amp = x+1 \\ y \amp = 2x-1 \\ y \amp = -x. \\ \end{aligned} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy all three equations?

in-context