Lesson 5: Solving Simultaneous Equations

👨🏽‍🏫 Introduction

Imagine you’re buying snacks: 2 doughnuts and 1 drink cost ₵10. Another time, 1 doughnut and 2 drinks cost ₵8. How much does each item cost?

When you have two unknowns and two equations at the same time, you’re dealing with simultaneous equations. These equations “work together” to help you find both unknown values.

By the end of this lesson, you’ll be able to:

• Understand what simultaneous equations are
• Solve them using the substitution or elimination method
• Check your solutions for accuracy

🧠 Key Concepts

📌 1. What Are Simultaneous Equations?

These are two (or more) equations with two unknowns, such as:

The goal is to find values of x and y that satisfy both equations at the same time.

🔄 2. Methods of Solving

🧩 Substitution Method

1. Make one variable the subject in one equation.
2. Substitute that expression into the second equation.
3. Solve for one variable, then substitute back to find the other.

⚖️ Elimination Method

1. Multiply one or both equations (if needed) so the coefficients of one variable are equal.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable, then substitute to find the other.

🧮 Sample Problem Walkthroughs

1. Solve:
   
   Using Elimination:
   Step 1: Add the equations:
   
   
   Step 2: Substitute x into the first equation:
   
2. Solve:
   
   Using Substitution:
   Step 1: Substitute x = y + 1 into the first equation:
   
   
   Step 2: x = y + 1 = 4.2

✍🏽 Practice Exercises

🅰️ A. Solve the following using any method:

✅ Answers and Explanations

• 1. Add: , then
• 2. Substitute x = 3 →
• 3. From 2nd equation: , substitute into 1st:
  

📌 Recap

• ✅ Simultaneous equations involve two unknowns and two equations
• ✅ Use elimination to remove a variable and simplify
• ✅ Use substitution when one variable is already isolated
• ✅ Always check your answers in both equations

💭 Reflection Prompt

Have you ever had to split costs with a friend where both of you bought different items? How could simultaneous equations help you fairly figure out who owes what?