Solving Systems of Linear Equations in Two Variables

Systems of Linear Equations

Introduction to Systems of Equations

A system of two linear equations in two unknowns consists of two equations that share two variables. The solution to such a system is an ordered pair that satisfies both equations simultaneously.

• System of Equations: A set of two or more equations with the same variables.

• Solution: An ordered pair (x, y) that makes both equations true.

• Some systems have exactly one solution, some have none, and some have infinitely many solutions.

Example: Determine whether (3, 2) is a solution to the following system:

• $x + 3y = 3$

• $4x + 3y = 6$

To check, substitute x = 3 and y = 2 into both equations and verify if both are true.

Methods for Solving Systems of Equations

There are three primary methods for solving systems of two linear equations in two variables:

1. Graphing Method

2. Substitution Method

3. Addition (also called Elimination) Method

Graphing Method

Each equation is graphed on the same coordinate plane. The point of intersection (if any) represents the solution to the system.

• Write each equation in slope-intercept form ($y = mx + b$) if possible.

• Graph both lines on the same axes.

• The intersection point is the solution (if the lines intersect).

• If the lines are parallel, there is no solution. If they coincide, there are infinitely many solutions.

Example: Graph $x + 3y = 3$ and $4x + 3y = 6$ to find their intersection.

Substitution Method

This method involves solving one equation for one variable and substituting that expression into the other equation.

1. Choose one of the equations and solve 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 the value found back into one of the original equations to find the other variable.

Example: Solve $x + 3y = 3$ for $x$: $x = 3 - 3y$. Substitute into $4x + 3y = 6$ and solve for $y$.

Addition (Elimination) Method

Also known as the Elimination Method, this approach involves adding or subtracting equations to eliminate one variable.

1. Arrange both equations in the form $ax + by = c$.

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

3. Add or subtract the equations to eliminate one variable.

4. Solve for the remaining variable.

5. Substitute this value into one of the original equations to solve for the other variable.

Example: Multiply $x + 3y = 3$ by 4 to get $4x + 12y = 12$. Subtract $4x + 3y = 6$ from this to eliminate $x$ and solve for $y$.

Comparison of Methods

Method	Main Steps	Best Used When
Graphing	Graph both equations; find intersection	Quick estimate or visual solution
Substitution	Solve for one variable; substitute	One equation is easily solved for a variable
Addition/Elimination	Combine equations to eliminate a variable	Coefficients are easily matched or opposites

Key Terms

• System of Equations: A set of equations with the same variables.

• Solution: An ordered pair that satisfies all equations in the system.

• Consistent System: A system with at least one solution.

• Inconsistent System: A system with no solution.

• Dependent System: A system with infinitely many solutions (the equations represent the same line).

• Independent System: A system with exactly one solution (the lines intersect at one point).