Algebra 4: Systems of Equations - Mathematics Platform

🎯 Learning Objectives

Understand what a system of equations is
Solve systems using graphing method
Apply substitution method
Use elimination method
Interpret solutions geometrically
Solve real-world problems with systems

1. Understanding Systems of Equations

🔗 What is a System of Equations?

A system of equations is two or more equations with the same variables. The solution is the point(s) where all equations are true simultaneously.

Example System:

y = 2x + 1

y = -x + 4

Solution: The point where both lines intersect: (1, 3)

Types of Solutions:

One Solution

Lines intersect at one point

No Solution

Lines are parallel

Infinite Solutions

Lines are identical

2. Solving by Graphing

📊 Graphical Solutions

Graph both equations on the same coordinate plane. The intersection point(s) are the solution(s).

📈 Interactive System Grapher

Equation 1: y = x +

Equation 2: y = x +

🎯 Graphing Practice

Solve this system by graphing:

y = x + 2

y = -x + 4

Intersection point: (

)

3. Substitution Method

🔄 How Substitution Works

Solve one equation for one variable
Substitute that expression into the other equation
Solve the resulting equation
Substitute back to find the other variable
Check your solution

Example: Substitution Method

x + y = 7

2x - y = 4

Step 1: Solve first equation for y: y = 7 - x

Step 2: Substitute into second equation: 2x - (7 - x) = 4

Step 3: Solve: 2x - 7 + x = 4 → 3x = 11 → x = 11/3 ≈ 3.667

Step 4: Substitute back: y = 7 - 11/3 = 10/3 ≈ 3.333

Solution: (11/3, 10/3) or approximately (3.67, 3.33)

🔄 Interactive Substitution

x + y =

4. Elimination Method

➕ How Elimination Works

Add or subtract equations to eliminate one variable, then solve for the remaining variable.

Key Tips:

Multiply equations to make coefficients equal
Add equations to eliminate a variable
Subtract equations to eliminate a variable
Always check your solution

Example: Elimination Method

3x + 2y = 11

2x - 2y = 2

Step 1: Add equations to eliminate y: (3x + 2y) + (2x - 2y) = 11 + 2

Step 2: Combine: 5x = 13 → x = 13/5 = 2.6

Step 3: Substitute x = 2.6 into first equation: 3(2.6) + 2y = 11

Step 4: 7.8 + 2y = 11 → 2y = 3.2 → y = 1.6

Solution: (2.6, 1.6)

➕ Interactive Elimination

x + y =

5. Real-World Applications

🌍 Practical Uses

Systems of equations solve real problems where multiple conditions must be met simultaneously.

💰 Business: Pricing Strategy

A store sells shirts for $15 and pants for $25. They made $400 from 25 items.

s + p = 25

15s + 25p = 400

Solution: 12 shirts, 13 pants

🍕 Food Mixtures

Mixing candies: 30¢ and 50¢ pieces. 20 pieces total worth $8.00.

x + y = 20

0.30x + 0.50y = 8.00

Solution: 12 cheap, 8 expensive pieces

🚗 Distance Problems

Two trains leave stations 300 miles apart, heading toward each other.

Train A: 60 mph, Train B: 40 mph

60t + 40t = 300

t = time until meeting

Solution: Meet after 2.5 hours

💼 Business Problem

A coffee shop sells lattes for $4 and muffins for $3. In one day, they sold 50 items and made $155.

How many of each did they sell?

Let L = number of lattes

Let M = number of muffins

L + M = 50

4L + 3M = 155

Lattes:

Muffins: