Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 5: Multi-Variable Linear Equations

Lesson 1: Introduction to Two-Variable Linear Equations

In this Grade 4 AoPS Introduction to Algebra lesson, students learn what two-variable linear equations are and how to identify and generate their solutions as ordered pairs. Working with equations like 2x − 3y = 7, students practice substitution and discover patterns that reveal why these equations have infinitely many solutions. This lesson builds foundational skills in solving for one variable in terms of another, preparing students for AMC 8 and AMC 10 problem-solving.

Section 1

Linear equation in two variables

Property

An equation of the form $Ax + By = C$ , where $A$ and $B$ are not both zero, is called a linear equation in two variables.

A linear equation is in standard form when it is written $Ax + By = C$ .

An ordered pair $(x, y)$ is a solution of the linear equation $Ax + By = C$ , if the equation is a true statement when the $x$ - and $y$ -values of the ordered pair are substituted into the equation.

Section 2

Solution to a linear equation

Property

An ordered pair $(x, y)$ is a solution to the linear equation $Ax + By = C$ , if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation. To find a solution, choose any value for one variable, substitute it into the equation, and solve for the other variable.

Examples

To check if $(3, 1)$ is a solution to $2x + y = 7$ , substitute the values: $2(3) + 1 = 6 + 1 = 7$ . Since $7 = 7$ , it is a solution.
Find a solution to $y = 3x + 1$ . Let's choose $x=2$ . Substitute it in: $y = 3(2) + 1 = 7$ . So, $(2, 7)$ is a solution.
Find a solution to $4x - 2y = 10$ . Let's choose $y=1$ . Substitute it in: $4x - 2(1) = 10$ , which means $4x = 12$ , so $x=3$ . The ordered pair $(3, 1)$ is a solution.

Explanation

A solution is an $(x, y)$ pair that makes the equation true. Think of it as a specific point that lies on the equation's line. Since the line is infinite, there are infinitely many solutions you can find for any linear equation.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Linear equation in two variables

Property

An equation of the form $Ax + By = C$ , where $A$ and $B$ are not both zero, is called a linear equation in two variables.

A linear equation is in standard form when it is written $Ax + By = C$ .

Section 2

Solution to a linear equation

Property

Examples

To check if $(3, 1)$ is a solution to $2x + y = 7$ , substitute the values: $2(3) + 1 = 6 + 1 = 7$ . Since $7 = 7$ , it is a solution.
Find a solution to $y = 3x + 1$ . Let's choose $x=2$ . Substitute it in: $y = 3(2) + 1 = 7$ . So, $(2, 7)$ is a solution.
Find a solution to $4x - 2y = 10$ . Let's choose $y=1$ . Substitute it in: $4x - 2(1) = 10$ , which means $4x = 12$ , so $x=3$ . The ordered pair $(3, 1)$ is a solution.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Linear equation in two variables

Property

An equation of the form $Ax + By = C$ , where $A$ and $B$ are not both zero, is called a linear equation in two variables.

A linear equation is in standard form when it is written $Ax + By = C$ .

Section 2

Solution to a linear equation

Property

Examples

To check if $(3, 1)$ is a solution to $2x + y = 7$ , substitute the values: $2(3) + 1 = 6 + 1 = 7$ . Since $7 = 7$ , it is a solution.
Find a solution to $y = 3x + 1$ . Let's choose $x=2$ . Substitute it in: $y = 3(2) + 1 = 7$ . So, $(2, 7)$ is a solution.
Find a solution to $4x - 2y = 10$ . Let's choose $y=1$ . Substitute it in: $4x - 2(1) = 10$ , which means $4x = 12$ , so $x=3$ . The ordered pair $(3, 1)$ is a solution.