Equations
Welcome to our lesson on equations! Today, we will learn about what equations are, how to solve them, and see some examples from everyday life. Equations are a fundamental part of mathematics and are used to express relationships between numbers and variables.
What is an Equation?
An equation is a mathematical statement that shows that two expressions are equal. It has two sides separated by an equal sign (=). For example:
\( 3 + 2 = 5 \)
In this equation, the left side (3 + 2) is equal to the right side (5).
Parts of an Equation
Equations have different parts:
- Left Side: The expression on the left of the equal sign.
- Right Side: The expression on the right of the equal sign.
- Equal Sign: The symbol (=) that shows both sides are equal.
Types of Equations
There are different types of equations, but we will focus on simple ones for now:
- Simple Equations: These have numbers and one variable. For example: \( x + 3 = 7 \)
- Linear Equations: These have variables raised to the power of 1. For example: \( 2x + 3 = 7 \)
Solving Simple Equations
To solve an equation means to find the value of the variable that makes the equation true. Let's look at some examples:
Example 1: Solving \( x + 3 = 7 \)
Step-by-step solution:
- Start with the equation: \( x + 3 = 7 \)
- Subtract 3 from both sides to isolate \( x \): \( x + 3 - 3 = 7 - 3 \)
- Simplify: \( x = 4 \)
So, the solution is \( x = 4 \).
Example 2: Solving \( 2x + 3 = 7 \)
Step-by-step solution:
- Start with the equation: \( 2x + 3 = 7 \)
- Subtract 3 from both sides: \( 2x + 3 - 3 = 7 - 3 \)
- Simplify: \( 2x = 4 \)
- Divide both sides by 2: \( \frac{2x}{2} = \frac{4}{2} \)
- Simplify: \( x = 2 \)
So, the solution is \( x = 2 \).
Example 3: Solving \( x - 5 = 10 \)
Step-by-step solution:
- Start with the equation: \( x - 5 = 10 \)
- Add 5 to both sides: \( x - 5 + 5 = 10 + 5 \)
- Simplify: \( x = 15 \)
So, the solution is \( x = 15 \).
Real-World Applications
Equations are used in many real-life situations. Here are a few examples:
- Shopping: If you buy 3 apples and each apple costs $2, you can use an equation to find the total cost: \( 3 \times 2 = 6 \) dollars.
- Travel: If you are driving at a speed of 60 miles per hour and you want to know how far you will travel in 2 hours, you can use the equation: \( 60 \times 2 = 120 \) miles.
- Cooking: If a recipe needs 2 cups of flour and you want to make half the recipe, you can use the equation: \( \frac{2}{2} = 1 \) cup of flour.
Summary
Today, we learned about equations. Here are the key points:
- An equation is a mathematical statement showing that two expressions are equal.
- Equations have a left side, a right side, and an equal sign.
- We can solve simple equations by isolating the variable.
- Equations are used in everyday life, such as shopping, traveling, and cooking.
Understanding equations helps us solve problems and make decisions in our daily lives. Keep practicing, and you will become more comfortable with equations!
