Systems of Equations

🔓 Systems of Equations: The Detective’s Toolkit

Ever played a game where you have to find two secret numbers at the same time? That’s exactly what solving systems of equations is!

🎯 What is a System of Equations?

Imagine you’re a detective. Someone gives you two clues:

• “I’m thinking of two numbers. They add up to 10.”
• “The first number is 2 more than the second.”

One clue alone isn’t enough. But two clues together? Now you can crack the case!

A system of equations = two or more equations that share the same unknown numbers.

🌟 Simple Example

x + y = 10      ← Clue 1
x - y = 2       ← Clue 2

The mystery numbers? x = 6 and y = 4. Check it:

• 6 + 4 = 10 ✓
• 6 - 4 = 2 ✓

Both clues are satisfied. Case closed!

📊 Solving by Graphing

What if you could see the answer? You can!

The Big Idea: Each equation is a line. Where the lines cross? That’s your answer!

graph TD
    A["Graph Line 1"] --> B["Graph Line 2"]
    B --> C{Where do they cross?}
    C --> D["That point is your solution!"]

🎨 How It Works

System:

y = 2x + 1
y = -x + 4

1. Draw the first line (goes up, crosses y-axis at 1)
2. Draw the second line (goes down, crosses y-axis at 4)
3. Find where they meet: (1, 3)

Check: Plug x=1, y=3 into both equations:

• 3 = 2(1) + 1 = 3 ✓
• 3 = -(1) + 4 = 3 ✓

The crossing point is the solution!

🔄 Substitution Method

Think of it like a secret code swap.

The Trick: Solve one equation for one variable. Then substitute it into the other!

📝 Step-by-Step Example

System:

y = 3x        ← Equation 1
x + y = 8     ← Equation 2

Step 1: Equation 1 already tells us y equals 3x.

Step 2: Replace y in Equation 2:

x + (3x) = 8
4x = 8
x = 2

Step 3: Find y using x = 2:

y = 3(2) = 6

Solution: x = 2, y = 6

Verify: 2 + 6 = 8 ✓ and 6 = 3(2) ✓

⚖️ Elimination Method

Make one variable disappear like magic!

The Trick: Add or subtract the equations to eliminate one variable.

🎩 Magic Example

System:

2x + y = 7
x - y = 2

Watch the Magic: Add both equations together!

  2x + y = 7
+  x - y = 2
─────────────
  3x + 0 = 9

The y’s canceled out! Now:

3x = 9
x = 3

Plug x = 3 back in:

2(3) + y = 7
y = 1

Solution: x = 3, y = 1

🔧 Sometimes You Multiply First

System:

3x + 2y = 12
x + 2y = 8

The y-coefficients match! Subtract:

  3x + 2y = 12
-  x + 2y = 8
─────────────
  2x = 4
  x = 2

Then: y = 3

🧊 Three-Variable Systems

Now you have THREE mystery numbers. You need THREE clues!

🎲 Real Example

x + y + z = 6
2x - y + z = 3
x + 2y - z = 3

Strategy: Use elimination twice to reduce to two variables, then solve.

Step 1: Eliminate z using equations 1 and 3:

  x + y + z = 6
+ x + 2y - z = 3
────────────────
  2x + 3y = 9

Step 2: Eliminate z using equations 2 and 3:

  2x - y + z = 3
+ x + 2y - z = 3
────────────────
  3x + y = 6

Step 3: Now solve:

2x + 3y = 9
3x + y = 6

From second: y = 6 - 3x

Substitute: 2x + 3(6 - 3x) = 9

2x + 18 - 9x = 9
-7x = -9
x = 9/7

Continue to find y and z!

🚨 Inconsistent and Dependent Systems

Not all systems have a nice, single answer!

❌ Inconsistent (No Solution)

x + y = 5
x + y = 7

Wait… the same thing can’t equal 5 AND 7!

Visual: Parallel lines that never meet.

graph TD
    A["Line 1"] --> B["Never touches"]
    C["Line 2"] --> B
    B --> D["No solution exists"]

♾️ Dependent (Infinite Solutions)

x + y = 4
2x + 2y = 8

The second equation is just the first one doubled!

Visual: They’re the same line. Every point on the line is a solution!

How to spot it: One equation is a multiple of the other.

🌍 Applications of Systems

This isn’t just math homework. It’s real life!

💰 Money Problems

Scenario: You have 15 coins (nickels and dimes) worth $1.10.

Let n = nickels, d = dimes.

n + d = 15              ← Total coins
5n + 10d = 110          ← Total cents

Solve: n = 8 nickels, d = 7 dimes

Verify: 8 + 7 = 15 ✓ and 40¢ + 70¢ = $1.10 ✓

🚗 Travel Problems

Scenario: Two cars start 300 miles apart, driving toward each other. Car A goes 50 mph, Car B goes 70 mph. When do they meet?

Let t = time in hours.

Distance by A: 50t Distance by B: 70t

50t + 70t = 300
120t = 300
t = 2.5 hours

🛒 Mixture Problems

Scenario: You mix 20% juice with 80% juice to make 50 liters of 40% juice.

Let x = liters of 20%, y = liters of 80%.

x + y = 50             ← Total volume
0.20x + 0.80y = 0.40(50)  ← Concentration

Solve: x = 33.3 liters, y = 16.7 liters

🎓 Your Detective Toolkit - Summary

🏆 Pro Tips

1. Check your work! Plug solutions into BOTH original equations.
2. No solution? The lines are parallel.
3. Infinite solutions? Same line, just disguised.
4. Three variables? Eliminate systematically.

You’ve cracked the code! Systems of equations are just detective puzzles waiting to be solved. Two clues, two unknowns, one solution. Now go find those mystery numbers! 🔍