Steps to solve one-step division equations for 8th graders

📚 Understanding One-Step Division Equations

A one-step division equation is an algebraic equation that can be solved in just one step by using division. These equations have the basic form:

$\frac{x}{a} = b$

Where:

• 🧮 $x$ is the variable we need to find.
• 🔢 $a$ is the number by which the variable is divided.
• 📊 $b$ is the result of the division.

Our goal is to isolate $x$ on one side of the equation to find its value.

📜 Historical Context

The concept of solving equations dates back to ancient civilizations, with early forms found in Babylonian and Egyptian mathematics. However, the symbolic notation we use today evolved over centuries. The use of letters to represent unknowns and the development of algebraic techniques were crucial steps in simplifying and generalizing problem-solving methods.

🔑 Key Principles for Solving

• ⚖️ The Golden Rule of Algebra: What you do to one side of the equation, you must do to the other to maintain balance.
• ➗ Isolating the Variable: To solve for $x$, we need to get it alone on one side of the equation. Since $x$ is being divided by $a$, we'll use the inverse operation, which is multiplication.
• ✖️ Multiplication Property of Equality: Multiplying both sides of the equation by the same number maintains the equality.

📝 Step-by-Step Solution

Let's solve the equation $\frac{x}{a} = b$:

1. Multiply both sides of the equation by $a$:

   $a \cdot \frac{x}{a} = a \cdot b$
2. Simplify:

   $x = a \cdot b$

Therefore, the solution for $x$ is $a \cdot b$.

💡 Real-World Examples

Example 1: Sharing Cookies

Imagine you're sharing cookies with friends. You divide a box of cookies equally among 4 friends, and each friend gets 3 cookies. How many cookies were in the box originally?

Equation: $\frac{x}{4} = 3$

Solution:

$x = 4 \cdot 3 = 12$

There were originally 12 cookies in the box.

Example 2: Dividing Pizza

A pizza is cut into equal slices. If each slice is $\frac{1}{6}$ of the pizza, and you have 2 slices, what fraction of the pizza do you have?

Equation: $\frac{x}{2} = \frac{1}{6}$

Solution:

$x = 2 \cdot \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

🧪 Practice Quiz

1. Solve: $\frac{x}{5} = 7$
2. Solve: $\frac{y}{3} = 9$
3. Solve: $\frac{z}{10} = 4$
4. Solve: $\frac{a}{2} = 11$
5. Solve: $\frac{b}{6} = 8$
6. Solve: $\frac{c}{4} = 12$
7. Solve: $\frac{d}{7} = 5$

Answers: 1) 35, 2) 27, 3) 40, 4) 22, 5) 48, 6) 48, 7) 35

✅ Conclusion

One-step division equations are a fundamental concept in algebra. By understanding the principle of inverse operations and maintaining balance in the equation, you can easily solve for the unknown variable. Keep practicing, and you'll become a pro at solving these equations!