MATHEMATICS CLASS NOTES: SIMPLE EQUATIONS (SOLVING ONE-VARIABLE LINEAR EQUATIONS)

Grade Level: JSS 1
Subject: Mathematics
Topic: Simple Equations: Solving One-Variable Linear Equations

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1. COMPREHENSIVE CORE CONCEPTS

What is a Simple Equation?

In Mathematics, an equation is a mathematical statement that shows that two expressions are equal. You can think of an equation as a sentence in a language where the "verb" is the equals sign (=). A one-variable linear equation is an equation where there is only one unknown letter (called a variable, like $x, y, \text{ or } p$) and that letter is not squared or cubed (it has a power of 1).

The main goal of solving an equation is to find the value of the variable that makes the statement true. This value is called the solution or the root of the equation. For example, in the equation $x + 5 = 12$, the solution is $x = 7$ because $7 + 5 = 12$ is a true statement.

The Principle of the Balance Scale

The most important rule in solving equations is the Balance Principle. Imagine a traditional market balance scale. If the scale is balanced and you add 1kg to the left side, you must add 1kg to the right side to keep it level.

Similarly, in Mathematics, an equation is like a balanced scale. Whatever operation you perform on one side of the equals sign, you must perform the exact same operation on the other side. This keeps the "balance" of the equation intact.

Solving by Inverse Operations

To find the value of a variable, we need to get the variable all by itself on one side of the equals sign. We do this by using Inverse Operations (opposite operations).

1. Addition and Subtraction are inverses: To remove a "+5", you subtract 5 from both sides. To remove a "-3", you add 3 to both sides.
2. Multiplication and Division are inverses: To remove a "multiply by 4", you divide both sides by 4. To remove a "divide by 2", you multiply both sides by 2.

Step-by-Step Examples

Example 1: Solving by Subtraction

Equation: $x + 8 = 20$

• Identify the goal: We want $x$ by itself. Currently, 8 is being added to it.
• Inverse action: Subtract 8 from both sides.
• Step 1: $x + 8 - 8 = 20 - 8$
• Step 2: $x = 12$
• Check: Does $12 + 8 = 20$? Yes!

Example 2: Solving by Addition

Equation: $y - 15 = 30$

• Identify the goal: We want $y$ by itself. Currently, 15 is being subtracted.
• Inverse action: Add 15 to both sides.
• Step 1: $y - 15 + 15 = 30 + 15$
• Step 2: $y = 45$

Example 3: Solving by Division

Equation: $4m = 24$

• Identify the goal: We want $m$ by itself. $4m$ means $4 \times m$.
• Inverse action: Divide both sides by 4.
• Step 1: $\frac{4m}{4} = \frac{24}{4}$
• Step 2: $m = 6$

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2. REAL-WORLD EXAMPLES

Equations aren't just for classrooms; we use them every day without realizing it!

• Scenario A: The Market Trip: Imagine you go to a market in Lagos with ₦2,000. You buy a bag of rice and receive ₦700 as change. How much did the rice cost?

  • Equation: Let $r$ be the price of rice. $r + 700 = 2,000$.
  • Solution: $r = 2,000 - 700 \rightarrow r = ₦1,300$.

• Scenario B: Sharing Snacks: You have a pack of biscuits to share among 5 friends. If each friend gets 4 biscuits and there are none left, how many biscuits were in the pack?

  • Equation: Let $b$ be the total biscuits. $\frac{b}{5} = 4$.
  • Solution: $b = 4 \times 5 \rightarrow b = 20$ biscuits.

• Scenario C: Transportation: A Keke Napep driver charges ₦150 per passenger. If he made ₦1,200 in one trip, how many passengers did he carry?

  • Equation: $150 \times p = 1,200$.
  • Solution: $p = \frac{1,200}{150} \rightarrow p = 8$ passengers.

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3. PRACTICAL APPLICATIONS: STEP-BY-STEP GUIDE

How to Translate Word Problems into Equations

Most real-life math comes in words, not numbers. Use these steps to solve them:

1. Identify the Unknown: What are you trying to find? Assign a letter to it (e.g., $x$ for "cost of bread").
2. Look for Keywords:
   • "Sum", "Added to", "Increased by" $\rightarrow$ +
   • "Difference", "Less than", "Decreased by" $\rightarrow$ -
   • "Product", "Times", "Twice" $\rightarrow$ $\times$
   • "Shared", "Quotient", "Per" $\rightarrow$ $\div$
   • "Is", "Result is", "Gives" $\rightarrow$ =
3. Write the Equation: Combine the numbers and the variable.
4. Solve: Use inverse operations.
5. Check: Does the answer make sense in the story?

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4. SUGGESTED HOME PROJECTS

Project: The DIY Clothes Hanger Balance

This project helps you visualize how equations stay balanced.

Materials Needed:

• A plastic or wire clothes hanger.
• Two identical small plastic cups or containers.
• String or yarn.
• A handful of identical items (e.g., dried beans, pebbles, or bottle caps).
• A marker and tape.

Procedure:

1. Build the Scale: Tie one cup to the left end of the hanger and the other cup to the right end using equal lengths of string.
2. Calibrate: Hang the hanger on a door handle. Ensure it sits level (is balanced).
3. The "Unknown" Challenge:
   • Place a small, light object (like a matchbox) in the left cup. This is your variable "$x$".
   • Add bottle caps to the right cup one by one until the hanger is balanced again.
   • If it took 5 caps to balance, your equation was $x = 5$.
4. Addition Practice: Put $x$ (the matchbox) and 2 beans in the left cup. Add beans to the right cup until balanced. If it takes 7 beans on the right, your equation is $x + 2 = 7$. To "solve," remove 2 beans from both sides. Does $x$ equal the remaining 5 beans?

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5. LIFE SKILLS INTEGRATION

Why Equations Matter for Your Future

• Financial Literacy: When you grow up, you will manage a salary. If you earn $S$ amount and your rent is $R$, you need to solve $S - R = \text{Savings}$ to plan your life.
• Time Management: If you have 3 hours of free time and you spend 45 minutes on chores, how much time is left for football? That’s $45 + t = 180$ minutes.
• Career Connections:
  • Engineers: Use equations to determine how much weight a bridge can hold.
  • Nurses: Use equations to calculate the correct dosage of medicine based on a patient's weight.
  • Computer Programmers: Write "if/then" equations to make video games and apps work.
  • Architects: Use linear equations to ensure the walls of a house are the right height and distance apart.

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6. ASSESSMENT THROUGH APPLICATION

Answer the following by setting up an equation first, then solving it.

1. The Tailor's Task: A tailor needs 25 meters of fabric to sew school uniforms. He already has 11 meters. How many more meters ($m$) does he need to buy?
2. The Poultry Farm: A farmer has some chickens. After selling 14 of them, he has 26 left. How many chickens ($c$) did he have at the start?
3. Savings Goal: Chidi wants to buy a football for ₦3,500. He saves ₦500 every week. For how many weeks ($w$) must he save to buy the ball?
4. Cooking Rice: A recipe says you need 3 cups of water for every cup of rice. If you used 12 cups of water, how many cups of rice ($r$) did you cook?

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7. STUDENT REFLECTION QUESTIONS

• Why do you think we call the letter in an equation a "variable"? (Hint: Think of the word "vary").
• If you add 10 to the left side of an equation but only add 5 to the right side, is it still an equation? Why or why not?
• Think about your morning routine today. Can you describe one part of it using a simple math equation? (Example: Time I woke up + Time spent bathing = Time I left for school).
• How does learning to solve for "x" help you become a better problem-solver in real life, even when you aren't using numbers?