Education Technology

# Expressions and Equations / Equations of the Form $$ax+by = c$$

Activity 13 of 18
In this lesson students investigate linear equations of the form $$ax+by = c$$. Students use coordinate grids to explore the “trade off” or “exchange” between the values of the two variables in order to maintain the constraint imposed by the constant c.

## Planning and Resources

Objectives
Students should be able to explain the meaning of a solution for a linear equation in two variables, recognize the solution graphs as a line, and identify unique solutions to a linear equation of the form $$ax+by = c$$

Vocabulary
linear equation
variables
constraint
solution
intercepts
linear combinations

Standard:

## Lesson Snapshot

#### Understanding

This lesson challenges students to explain the meaning of a solution for a linear combination of two variables with a given constraint and to recognize the solution graphs as a line.

### What to look for

Students may note that for equations of the form $$ax+by$$ = c where $$a$$ and $$b$$ are positive, the slope of the "solution line" is always negative.

### Sample Assessment

Which of the following ordered pairs $$(x, y)$$ is a solution to the equation $$2x+3y = 6$$?

a. (6, 3)
b. (3, 0)
c. (3, 2)
d. (2, 3)
e. (0, 3)

#### The Big Idea

Develop linear combinations of two quantities and investigate what happens when a constraint is imposed on the linear combination ax+by for a given a and b.

### What are the students doing?

Students explore a variety of contexts for a linear combination of two quantities with a given constraint.

### What is the teacher doing?

Encourage students to explore the exchange between the two quantities and how that exchange maps out the solution on a coordinate grid.