Learn on PengienVision, Algebra 1Chapter 4: Systems of Linear Equations and Inequalities

Lesson 1: Solving Systems of Equations by Graphing

In this Grade 11 enVision Algebra 1 lesson, students learn how to solve systems of linear equations by graphing, identifying the point of intersection as the solution. The lesson covers three possible outcomes — one solution, infinitely many solutions, and no solution — and applies graphing techniques to real-world problems such as comparing reading rates or ATV travel distances. Students also use graphing utilities to find approximate solutions when intersection points fall between integer values.

Section 1

Defining a System of Linear Equations

Property

A system of linear equations is a set of two or more linear equations that share the same variables. A solution to the system is an ordered pair $(x, y)$ that satisfies every equation in the system. The general form for a system of two linear equations is:

\begin{cases} A_1x + B_1y = C_1 \\ A_2x + B_2y = C_2 \end{cases}

Examples

A system with variables $x$ and $y$ :

\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

A system in standard form:

\begin{cases} 3x - 2y = 8 \\ x + 5y = -3 \end{cases}

Explanation

A system of linear equations consists of multiple linear equations considered together. The goal is to find a single point, or a set of points, that makes all equations in the system true simultaneously. Graphically, this corresponds to finding the point(s) where the lines representing each equation intersect. The brace symbol is used to group the equations together, indicating they form a system.

Section 2

The Three Possible Outcomes

Property

A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically:

One Solution (Intersecting): The lines have different slopes. The system is consistent and independent.
No Solution (Parallel): The lines have the exact same slope but different y-intercepts. The system is inconsistent.
Infinite Solutions (Coincident): The lines have the same slope and the same y-intercept. The system is consistent and dependent.

Examples

One Solution: The lines $y = x + 1$ and $y = -x + 3$ intersect at the point $(1, 2)$ , giving exactly one solution.
No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y-intercepts) and never intersect.
Infinite Solutions: The equations $y = 3x - 2$ and $6x - 2y = 4$ represent the same line when graphed, so every point on the line is a solution.

Explanation

When you graph two lines, they can only relate in three ways: they cross once, they never cross because they are parallel, or they are actually the exact same line. By simply looking at the 'm' and 'b' in their equations, you can instantly predict how many solutions the system has without even needing to draw the graph!

Section 3

Classifying Systems of Equations

Property

Systems of linear equations are classified by the number of solutions they have, which can be predicted by comparing their slopes ( $m$ ) and y-intercepts ( $b$ ) in slope-intercept form ( $y = mx + b$ ):

Consistent and Independent (1 Solution): The lines have different slopes. They intersect at exactly one point.
Inconsistent (No Solution): The lines have the same slope but different y-intercepts. They are parallel lines that never intersect.
Consistent and Dependent (Infinite Solutions): The lines have the same slope and the same y-intercept. They are coincident (the exact same line), meaning they intersect everywhere.

Examples

One Solution (Independent): The system $y = 4x + 1$ and $y = 2x + 3$ has different slopes ( $4$ and $2$ ). The lines will cross exactly once, at the point $(1, 5)$ .
No Solution (Inconsistent): The system $y = 3x + 4$ and $y = 3x - 2$ has the same slope ( $3$ ) but different y-intercepts ( $4$ and $-2$ ). The lines are parallel and will never touch.
Infinite Solutions (Dependent): The system $x + y = 5$ and $3x + 3y = 15$ . If you simplify the second equation by dividing everything by 3, you get $x + y = 5$ . They represent the exact same line, so every point on the line is a solution.

Explanation

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining a System of Linear Equations

Property

\begin{cases} A_1x + B_1y = C_1 \\ A_2x + B_2y = C_2 \end{cases}

Examples

A system with variables $x$ and $y$ :

\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

A system in standard form:

\begin{cases} 3x - 2y = 8 \\ x + 5y = -3 \end{cases}

Explanation

Section 2

The Three Possible Outcomes

Property

A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically:

One Solution (Intersecting): The lines have different slopes. The system is consistent and independent.
No Solution (Parallel): The lines have the exact same slope but different y-intercepts. The system is inconsistent.
Infinite Solutions (Coincident): The lines have the same slope and the same y-intercept. The system is consistent and dependent.

Examples

One Solution: The lines $y = x + 1$ and $y = -x + 3$ intersect at the point $(1, 2)$ , giving exactly one solution.
No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y-intercepts) and never intersect.
Infinite Solutions: The equations $y = 3x - 2$ and $6x - 2y = 4$ represent the same line when graphed, so every point on the line is a solution.

Explanation

Section 3

Classifying Systems of Equations

Property

Consistent and Independent (1 Solution): The lines have different slopes. They intersect at exactly one point.
Inconsistent (No Solution): The lines have the same slope but different y-intercepts. They are parallel lines that never intersect.
Consistent and Dependent (Infinite Solutions): The lines have the same slope and the same y-intercept. They are coincident (the exact same line), meaning they intersect everywhere.

Examples

One Solution (Independent): The system $y = 4x + 1$ and $y = 2x + 3$ has different slopes ( $4$ and $2$ ). The lines will cross exactly once, at the point $(1, 5)$ .
No Solution (Inconsistent): The system $y = 3x + 4$ and $y = 3x - 2$ has the same slope ( $3$ ) but different y-intercepts ( $4$ and $-2$ ). The lines are parallel and will never touch.
Infinite Solutions (Dependent): The system $x + y = 5$ and $3x + 3y = 15$ . If you simplify the second equation by dividing everything by 3, you get $x + y = 5$ . They represent the exact same line, so every point on the line is a solution.

Explanation

Overview

Lesson overview

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

Overview

Section 1

Defining a System of Linear Equations

Property

\begin{cases} A_1x + B_1y = C_1 \\ A_2x + B_2y = C_2 \end{cases}

Examples

A system with variables $x$ and $y$ :

\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

A system in standard form:

\begin{cases} 3x - 2y = 8 \\ x + 5y = -3 \end{cases}

Explanation

Section 2

The Three Possible Outcomes

Property

A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically:

One Solution (Intersecting): The lines have different slopes. The system is consistent and independent.
No Solution (Parallel): The lines have the exact same slope but different y-intercepts. The system is inconsistent.
Infinite Solutions (Coincident): The lines have the same slope and the same y-intercept. The system is consistent and dependent.

Examples

One Solution: The lines $y = x + 1$ and $y = -x + 3$ intersect at the point $(1, 2)$ , giving exactly one solution.
No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y-intercepts) and never intersect.
Infinite Solutions: The equations $y = 3x - 2$ and $6x - 2y = 4$ represent the same line when graphed, so every point on the line is a solution.

Explanation

Section 3

Classifying Systems of Equations

Property

Consistent and Independent (1 Solution): The lines have different slopes. They intersect at exactly one point.
Inconsistent (No Solution): The lines have the same slope but different y-intercepts. They are parallel lines that never intersect.
Consistent and Dependent (Infinite Solutions): The lines have the same slope and the same y-intercept. They are coincident (the exact same line), meaning they intersect everywhere.

Examples

One Solution (Independent): The system $y = 4x + 1$ and $y = 2x + 3$ has different slopes ( $4$ and $2$ ). The lines will cross exactly once, at the point $(1, 5)$ .
No Solution (Inconsistent): The system $y = 3x + 4$ and $y = 3x - 2$ has the same slope ( $3$ ) but different y-intercepts ( $4$ and $-2$ ). The lines are parallel and will never touch.
Infinite Solutions (Dependent): The system $x + y = 5$ and $3x + 3y = 15$ . If you simplify the second equation by dividing everything by 3, you get $x + y = 5$ . They represent the exact same line, so every point on the line is a solution.