# Lesson 11

On Both of the Lines

### Lesson Narrative

For the next several lessons, students will study systems of linear equations where the context is of the distance-versus-time variety, where there is an initial value and a rate of change. The equations in the system are in the form $$y = mx + b$$. Such contexts are useful in thinking about the meaning of the solution to the system (the time when two quantities are equal). The purpose of this lesson is to introduce students to the graphical interpretation of such systems. Keeping the graphs in mind will be useful as students navigate algebraic techniques for solving systems in the lessons to come.

The first activity after the warm-up focuses on the solution by drawing a graph for a ladybug's motion, marking the point on the graph where an ant and a ladybug meet, and asking the student to fill in the graph of the ant. The second activity draws attention to systems which have infinitely many solutions because the graphs of the equations are identical. This is interpreted as two runners staying together for the entire duration of a race.

### Learning Goals

Teacher Facing

• Create a graph that represents two linear relationships in context, and interpret (orally and in writing) the point of intersection.
• Interpret a graph of two equivalent lines in context.

### Student Facing

Let’s use lines to think about situations.

### Student Facing

• I can use graphs to find an ordered pair that two real-world situations have in common.