📋 Prerequisites

• Linear Equations & Inequalities

When you have two variables, you need two equations to find both values. Systems of equations show up everywhere — coordinate geometry, word problems, even number theory. Master substitution and elimination, and you'll solve 90% of competition system problems.

📚 Key Concepts

What Is a System?

A system of equations is a set of equations that share the same variables. A solution is a pair (or tuple) that makes all equations true at the same time.

For two linear equations with two variables, there are three possibilities:

• One solution: Lines intersect at one point (different slopes)
• No solution: Lines are parallel (same slope, different intercepts)
• Infinite solutions: Same line (same slope and intercept)

Method 1: Substitution

Best when one equation already has a variable isolated, or can be easily isolated.

1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve the resulting single-variable equation
4. Plug back to find the other variable

Example: $ y = 2x + 1 $ and $ 3x + y = 11 $ \(\to\) substitute: $ 3x + (2x+1) = 11 $ \(\to\) $ x = 2 $, $ y = 5 $

Method 2: Elimination (Addition)

Best when coefficients line up nicely, or when substitution would create fractions.

1. Line up equations with like terms stacked
2. Multiply one or both equations so one variable has opposite coefficients
3. Add the equations to eliminate that variable
4. Solve, then plug back to find the other variable

Key idea: if $ax + by = c$ and $dx + ey = f$, you can scale to make the $x$ (or $y$) coefficients cancel.

\(Three Variables (3 \\times 3 Systems)\)

Same elimination idea, just more steps:

1. \(Take two equations, eliminate one variable \\to get a 2-variable equation\)
2. \(Take a different pair, eliminate the same variable \\to get another 2-variable equation\)
3. \(Now you have a 2 \\times 2 system\) — solve with substitution or elimination
4. Plug back to find the third variable

Always check your answer in all three original equations!

Word Problems with Systems

Most system word problems fall into these categories:

• Two quantities with two totals: e.g., tickets: adult + child, total people + total money
• Mixture problems: two solutions mixed to get a target concentration
• Distance / rate: two objects, or one object with two different speeds
• Number digit problems: tens digit + units digit, value of number vs reversed

✏️ Example Problems