Education Technology

Expressions and Equations / Equations of the Form \(ax+by = c\)

Grade Level 8
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

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 \(ax+by = c\)

linear equation
linear combinations

Standard: Search Standards Alignment


Lesson Snapshot


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)

Answer: b

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.