Section 4.1 Systems of Linear Equations in Two Variables

Introduction In this section we will explore systems of linear equations and their solutions. A system of linear equations is in the form

Solutions We know from Chapter 4 that the graph of every linear equation is a straight line. When you have two linear equations, you have (at most) two lines. There are three possible scenarios for the relationship between those lines:

1. The lines intersect in a single point The system will have one ordered pair solution.

2. The two lines are parallel. There are no ordered pair solutions.

3. The two lines are actually the same line. There are infinitely many ordered pair solutions.

Solving Methods 1.Substitution One of the equations has an isolated variable, or a variable that can be easily isolated. Substitute what the variable is equal to into the other equation. Solve the resulting equation. Use that solution to find the other variable.

Examples

Solving Methods 2.Elimination Put both equations into standard form. If necessary, multiply one or both equations by some number(s) to create a set of opposite coefficients. Add the equations together. One variable will cancel. Solve for the remaining variable. Substitute into either equation to find the other variable.

Examples

Special Cases Both variables cancel out. If the resulting statement is true, you have infinitely many solutions (the two equations make the same line). If the resulting statement is false, you have no solution (the two equations make parallel lines).

Examples