Systems of Linear Equations in Two Variables (Chapter 4, Section 4.1)

Systems of Linear Equations and Inequalities

Introduction

This section introduces the concept of systems of linear equations in two variables, a foundational topic in College Algebra. Students learn how to determine solutions to systems, classify systems based on their solutions, and apply graphical and algebraic methods for solving.

Systems of Linear Equations in Two Variables

Definition and Solution

• System of Linear Equations: A set of two or more linear equations considered simultaneously, typically in the form: $\begin{cases} y = m_1x + b_1 \\ y = m_2x + b_2 \end{cases}$

• Solution to a System: An ordered pair $(x, y)$ that satisfies all equations in the system.

• Checking Solutions: Substitute the values of $x$ and $y$ into each equation. If both equations are true, the pair is a solution.

• Example: For the system $y = -4x$ and $y = -2x + 8$, check if $(-4, 16)$ is a solution:

  • $y = -4x \rightarrow 16 = -4(-4) = 16$

  • $y = -2x + 8 \rightarrow 16 = -2(-4) + 8 = 8 + 8 = 16$

  • Both equations are satisfied, so $(-4, 16)$ is a solution.

Graphical Solutions

Systems of equations can be solved by graphing each equation and identifying the point(s) of intersection.

• Consistent System: The lines intersect at a single point; there is one unique solution.

• Inconsistent System: The lines are parallel and do not intersect; there is no solution.

• Dependent System: The lines coincide (are the same line); there are infinitely many solutions.

Graphical Examples

• Consistent System Example:

  • Equations: $y = -3x - 6$ and $y = 2x - 1$

  • Graph: Lines intersect at $(-1, -3)$

  • Check: Substitute $x = -1$, $y = -3$ into both equations to verify.

• Inconsistent System Example:

  • Equations: $3x - 2y = -4$ and $9x + 6y = -6$

  • Graph: Lines are parallel; no intersection.

  • No solution exists.

• Dependent System Example:

  • Equations: $4x - 6y = 8$ and $-2x + 3y = -4$

  • Graph: Lines coincide; every point on one line is also on the other.

  • Infinitely many solutions.

Methods for Solving Systems of Linear Equations

Substitution Method

The substitution method involves solving one equation for one variable and substituting this expression into the other equation.

1. Solve one equation for one variable in terms of the other.

2. Substitute this expression into the other equation.

3. Solve the resulting equation for the remaining variable.

4. Substitute back to find the value of the first variable.

5. Check the solution in both original equations.

Example: Solve $2x - y = 13$ and $-4x - 9y = 7$ by substitution.

• Solve $2x - y = 13$ for $y$: $y = 2x - 13$

• Substitute into $-4x - 9y = 7$: $-4x - 9(2x - 13) = 7$

• Simplify: $-4x - 18x + 117 = 7 \rightarrow -22x = -110 \rightarrow x = 5$

• Substitute $x = 5$ into $y = 2x - 13$: $y = 10 - 13 = -3$

• Solution: $(5, -3)$

Addition (Elimination) Method

The addition method (also called elimination) involves adding or subtracting equations to eliminate one variable.

1. Arrange equations in standard form: $ax + by = c$

2. Multiply one or both equations by suitable numbers so that the coefficients of one variable are opposites.

3. Add the equations to eliminate one variable.

4. Solve for the remaining variable.

5. Substitute back to find the other variable.

6. Check the solution in both original equations.

Example: Solve $-x + y = 2$ and $x + y = 4$ by addition.

• Add: $(-x + y) + (x + y) = 2 + 4 \rightarrow 2y = 6 \rightarrow y = 3$

• Substitute $y = 3$ into $x + y = 4$: $x + 3 = 4 \rightarrow x = 1$

• Solution: $(1, 3)$

Classification of Systems

Types of Solutions

Type	Graphical Interpretation	Algebraic Interpretation
One unique solution	Lines intersect at one point	Obtain one value for $x$ and one for $y$
No solution	Lines are parallel	Obtain a false statement (e.g., $21 = 12$)
Infinite solutions	Lines coincide	Obtain a true statement (e.g., $0 = 0$)

Choosing an Appropriate Method

• Substitution: Best when one variable has a coefficient of 1 or -1.

• Addition: Useful when variables have matching or opposite coefficients, or when equations contain decimals or fractions.

Summary Table: System Types

System Type	Number of Solutions	Graphical Representation	Algebraic Result
Consistent	One	Lines intersect	Unique solution $(x, y)$
Inconsistent	None	Lines are parallel	Contradictory statement
Dependent	Infinite	Lines coincide	Identity (always true)

Additional info: These notes are based on textbook slides and cover the essential methods and classifications for solving systems of linear equations in two variables, as required for College Algebra.