Systems of Linear Equations and Inequalities

Introduction to Systems of Linear Equations

Systems of linear equations consist of two or more equations of the form Ax + By = C. When graphed, each equation represents a straight line. The solution to a system is an ordered pair that satisfies all equations simultaneously. Systems can have one solution, no solution, or infinitely many solutions.

• Linear System: A set of two or more linear equations involving the same variables.

• Solution: An ordered pair (x, y) that makes all equations in the system true.

• Types of Solutions:

  • Exactly one solution (lines intersect at a single point)

  • No solution (lines are parallel and never intersect)

  • Infinitely many solutions (lines coincide)

Determining Whether an Ordered Pair Is a Solution

To check if an ordered pair is a solution to a system, substitute the values of x and y into each equation. If both equations are satisfied, the pair is a solution.

• Step 1: Substitute x and y into each equation.

• Step 2: Check if both equations yield true statements.

• Example: For the system:

  • Equation 1:

  • Equation 2:

  Substitute (7, 6): If either equation is not satisfied, (7, 6) is not a solution.

Solving Systems by Graphing

Graphing is a visual method to solve systems of linear equations. The intersection point of the lines represents the solution.

• Step 1: Graph the first equation using intercepts or slope-intercept form.

• Step 2: Graph the second equation on the same axes.

• Step 3: Identify the intersection point; this is the solution.

• Step 4: Check the solution in both equations.

• Example: Solve by graphing:

  • Equation 1:

  • Equation 2:

  Find x- and y-intercepts for each equation, graph, and locate the intersection.

Number of Solutions to a System

The number of solutions depends on the relationship between the lines:

• One Solution: Lines intersect at a single point.

• No Solution: Lines are parallel (same slope, different y-intercepts).

• Infinitely Many Solutions: Lines coincide (same slope and y-intercept).

Example: System with No Solution

If two lines have the same slope but different y-intercepts, they are parallel and never intersect. The system is inconsistent and has no solution.

• Example: and

• Solution Set: The empty set,

Example: System with Infinitely Many Solutions

If two equations are different forms of the same line, every point on the line is a solution. These are called dependent equations.

• Example: and

• Solution Set: All points on the line

Using Graphs to Solve Real-World Problems

Graphing systems can help visualize and solve practical problems, such as comparing costs or modeling situations.

• Example: Toll bridge costs:

  • Without discount:

  • With discount pass:

• Graph both equations. The intersection point (10, 20) means using the bridge 10 times costs $20 with or without the discount pass.

Summary Table: Types of Solutions in Linear Systems