What is a system of equations?

A system of equations is two or more equations with the same variables that are considered together.

• Goal: Find the values of variables that satisfy all equations at the same time.

• Usually involves x and y in algebra 1.

Example:

• The solution is the point (x, y) where both lines intersect.

Why systems of equations matter

• Represent real-world problems with multiple conditions:

  • Budget problems

  • Speed and distance problems

  • Mixing solutions or ingredients

• Helps you find where two conditions are true at the same time.

Methods to solve systems of equations

There are three main methods:

1️⃣ Graphing

• Draw both equations on a coordinate plane

• Look for the intersection point

• That point (x, y) is the solution

Example:

• Graph both lines

• Intersection at (2, 4)

• ✅ Solution: (2, 4)

2️⃣ Substitution

• Solve one equation for one variable, then substitute into the other equation

Example:

Step 1: Substitute y from Eq1 into Eq2

Step 2: Substitute x into Eq1

✅ Solution: (2, 5)

3️⃣ Elimination (Addition/Subtraction)

• Add or subtract equations to eliminate one variable

Example:

Step 1: Add Eq1 and Eq2

Step 2: Substitute x into Eq2

✅ Solution: (10/3, 7/3)

Special cases

1. No solution

   • Lines are parallel (same slope, different intercepts)

2. Infinite solutions

   • Lines are the same (overlap completely)

Graphing reminder

• Intersection → solution

• Parallel lines → no solution

• Same line → infinite solutions

Common beginner mistakes

1. ❌ Forgetting to substitute correctly

2. ❌ Mistakes with adding/subtracting fractions

3. ❌ Forgetting to flip inequality sign if combining with inequalities

4. ❌ Misreading graph intersections

Why systems of equations matter

• Solve real-world problems with multiple variables

• Foundation for linear programming and algebra 2 topics

• Helps develop critical thinking and problem-solving skills