Lesson 17.1

Systems of Linear Equations in Two Variables

When one equation isn't enough—finding the point where two lines meet.

Introduction

A single linear equation has infinitely many solutions—a whole line of them! But when we need to find one specific point, we need a second equation. A system of equations asks: where do two lines intersect? Using substitution andelimination, we can find that unique point—or discover the lines are parallel (no solution) or identical (infinitely many).

Prerequisite Connection

You can solve single linear equations and graph lines in form.

Today's Increment

We combine TWO equations to find their common solution using substitution and elimination.

Why This Matters

Systems appear in optimization, physics, and economics. In calculus, we solve systems to find critical points.

Key Concepts

Definition

A system of linear equations consists of two or more equations. A solution is an ordered pair that satisfies ALL equations simultaneously.

Substitution Method

Solve one equation for one variable

Substitute into the other equation

Solve for the remaining variable

Back-substitute to find the other

Elimination Method

Multiply to get opposite coefficients

Add equations to eliminate a variable

Solve for the remaining variable

Substitute back to find the other

Worked Examples

Example 1: Substitution Method (Basic)

Solve the system:

Step 1: Substitute

Step 2: Solve for

Step 3: Find

Solution:

Example 2: Elimination Method (Intermediate)

Solve the system:

Step 1: Add equations (y-terms cancel)

Step 2: Solve for

Step 3: Substitute back

Solution:

Example 3: Special Cases (Advanced)

No Solution (Parallel Lines)

Same slope, different intercepts → never intersect

Contradiction → No solution

Infinitely Many (Same Line)

Second equation is just 2× the first

Always true → Infinitely many solutions

Common Pitfalls

Sign errors in elimination

When multiplying an equation by a negative, you must negate ALL terms, not just the variable you're targeting.

Forgetting to back-substitute

After finding one variable, always substitute it into one of the ORIGINAL equations to find the other.

Not checking the solution

Always verify your answer by plugging it into BOTH original equations.

Real-World Application

Break-Even Analysis in Business

Companies use systems of equations to find the break-even point—where revenue equals cost.

If cost = and revenue = , setting gives , so units to break even.