Systems of Equations: Geometry and Graphing

In this lesson:

• Understand what a "solution to a system" means geometrically
• Classify systems by number of solutions using slopes
• Solve systems by graphing and verify algebraically

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Interpret a system's solution as the intersection of two lines
2. Identify the three solution types and their geometric meaning
3. Graph a system and estimate the intersection point
4. Verify a solution by substituting into both equations

What Does "Solving a System" Mean?

A system is two equations that must be true simultaneously.

A solution is an ordered pair satisfying both equations.

Geometrically: each equation is a line — a solution is the intersection point.

How many times can two lines intersect?

The Three Geometric Cases for Systems

1. Different slopes → lines intersect once → one solution
2. Same slope, different intercepts → parallel lines → no solution
3. Same slope, same intercept → same line → infinitely many solutions

Case 1: One Solution — Different Slopes

Example: and

• Slopes: and — different → lines will intersect

We'll find the intersection at using algebra later. For now: different slopes guarantee exactly one solution.

Case 2: No Solution — Parallel Lines

Example: and

• Same slope ; different intercepts → parallel

These lines never meet. No solution.

Algebraic signal: a false statement like .

Case 3: Infinitely Many Solutions

Example: and

Rewrite: — identical equations

Every point on the line is a solution.

Algebraic signal: a true identity like .

Quick Check: Predict the Number of Solutions

How many solutions — without solving?

1. and
2. and
3. and

Compare slopes and intercepts

Solving Systems Using the Graphing Method

Best for: integer solutions and visual interpretation.

Steps:

1. Rewrite each equation in slope-intercept form
2. Graph using y-intercept and slope
3. Identify the intersection
4. Verify in both original equations

Limitation: approximate for non-integer solutions.

Graphing Method: Setting Up the Example

System: and

Convert to slope-intercept form:

Different slopes → exactly one solution exists

Graphing Example: Finding the Solution

The lines appear to intersect at .

Verify in both equations: ✓ and ✓

Solution:

Understanding the Limitations of Graphing

Exact: integer or simple fraction solutions — readable from a graph.

Approximate: decimal or irrational solutions — graph is insufficient.

Example: and

Exact solution: — use algebra (Lesson 2).

Practice Problems: Graphing Linear Systems

Predict solutions, then graph to confirm:

1. and
2. and

Problem 1: graph and verify algebraically.
Problem 2: rewrite the second equation first.

Answers to Graphing Linear Systems Practice

1. Different slopes → one solution at .
   Verify: ✓; ✓

2. Rewrite: — same as equation 1.
   Infinitely many solutions — same line.

Key Takeaways: Geometry and Graphing

Three cases:

• Different slopes → one solution (lines intersect)
• Same slope, different intercepts → no solution (parallel)
• Proportional equations → infinitely many (coincident)

Graphing method: convert to slope-intercept → graph → read intersection → verify in both equations

Graphing limitation: approximate for non-integer solutions

What's Next: Substitution and Elimination

Lesson 2: Substitution and Elimination Methods

• Substitution: solve one equation for one variable, plug into the other
• Elimination: scale equations so one variable cancels when added

These methods give exact solutions — including fractions and decimals.