Learn on PengiBig Ideas Math, Course 1Chapter 7: Equations and Inequalities

Lesson 4: Writing Equations in Two Variables

In this Grade 6 lesson from Big Ideas Math, Course 1 (Chapter 7), students learn how to write equations in two variables by exploring real-world situations such as hourly earnings and perimeter formulas. Students identify independent and dependent variables, find solutions of two-variable equations as ordered pairs, and use tables and graphs to analyze the relationship between two related quantities. This lesson addresses Common Core standard 6.EE.9.

Section 1

Identifying Independent and Dependent Variables

Property

In a relationship between two quantities, the independent variable is the quantity that is changed or controlled (the cause).
The dependent variable is the quantity that is measured or observed as a result (the effect).

Examples

Section 2

Writing Equations in Two Variables

Property

An equation in two variables shows the relationship between an independent variable (input) and a dependent variable (output).
These equations have the general form where one variable is expressed in terms of another, such as y=mx+by = mx + b, y=kxy = kx, or other forms like A=lwA = lw. In these equations:

  • One variable represents the input (independent variable)
  • The other variable represents the output (dependent variable)
  • Constants and coefficients determine how the variables relate to each other

Examples

Section 3

Understanding Solutions of Two-Variable Equations

Property

An equation in two variables, such as y=4x−2y = 4x - 2, has many solutions.
Each solution consists of an ordered pair of values, one for xx and one for yy, that together satisfy the equation (make the equation true.)

Examples

The ordered pair (1,3)(1, 3) is not a solution of y=4x−2y = 4x - 2 because substituting the values gives 4(1)−2=24(1) - 2 = 2. Since the y-value is 3, and 3≠23 \ne 2, the pair is not a solution.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Identifying Independent and Dependent Variables

Property

In a relationship between two quantities, the independent variable is the quantity that is changed or controlled (the cause).
The dependent variable is the quantity that is measured or observed as a result (the effect).

Examples

Section 2

Writing Equations in Two Variables

Property

An equation in two variables shows the relationship between an independent variable (input) and a dependent variable (output).
These equations have the general form where one variable is expressed in terms of another, such as y=mx+by = mx + b, y=kxy = kx, or other forms like A=lwA = lw. In these equations:

  • One variable represents the input (independent variable)
  • The other variable represents the output (dependent variable)
  • Constants and coefficients determine how the variables relate to each other

Examples

Section 3

Understanding Solutions of Two-Variable Equations

Property

An equation in two variables, such as y=4x−2y = 4x - 2, has many solutions.
Each solution consists of an ordered pair of values, one for xx and one for yy, that together satisfy the equation (make the equation true.)

Examples

The ordered pair (1,3)(1, 3) is not a solution of y=4x−2y = 4x - 2 because substituting the values gives 4(1)−2=24(1) - 2 = 2. Since the y-value is 3, and 3≠23 \ne 2, the pair is not a solution.