Understanding Equations

In algebra, an equation is like a balanced scale. It is a mathematical statement that shows two expressions are equal.

An equation uses an equal sign (=) to indicate that what’s on the left side has the same value as what’s on the right side. For example:

3 + 2 = 5

This equation tells us that adding 3 and 2 gives the same result as 5. Let’s dive deeper to understand what equations are, how they work, and why they matter.

Features of an Equation

  • Balance

    An equation maintains equality between its two sides, just like a balanced scale.

  • Solvable

    Every equation has a solution, which involves finding the value(s) of the variable(s) that make it true.

  • Operations

    Equations include mathematical operations such as addition, subtraction, multiplication, or division.

Equations Matter

Equations are the backbone of algebra. They help us:

  • Solve Real World Problems: Equations help us find answers to practical issues, like calculating expenses, determining travel time, or dividing resources fairly.

  • Predict Outcomes and Make Decisions: By using equations, we can forecast future results, such as predicting earnings based on hours worked or estimating savings over time.

  • Understand Patterns and Relationships: Equations reveal how different quantities are connected, such as the relationship between speed, distance, and time, or supply and demand in economics.

Types of Equations

In Algebra, you will be faced with different types of equations, and they can be solved in different ways.

  • Linear Equations: These involve variables raised to the power of 1. For example the equation: 2x + 5 = 9.

  • Quadratic Equation: These include variables raised to the power of 2. For example the equation: x² + 3x + 2 = 0.

Solving Equations Step by Step

Let’s look at how to solve different kinds of equations.

Example 1: Simple Equation

Solve x + 5 = 12:

  1. Subtract 5 from both sides: x = 12 - 5

  2. Simplify: x = 7

Example 2: Multi-Step Equation

Solve 3x - 4 = 14:

  1. Add 4 to both sides: 3x = 18

  2. Divide by 3: x = 6

Example 3: Equation with Fractions

Solve (x/2) + 3 = 8:

  1. Subtract 3 from both sides: x/2 = 5

  2. Multiply both sides by 2: x = 10

Real World Equations

Equations are everywhere in real life.

Example 1: Shopping

Imagine you have $50, and you need to buy groceries. After shopping, you have $20 left. How much money did you spend? To figure it out, you can use the equation:

50 - x = 20

Here, x represents how much money you spent. Solving for x:

x = 50 - 20 = 30

This means you spent $30 shopping. Equations like this are helpful for tracking expenses and staying within budget.

Example 2: Traveling

Suppose you’re driving at a constant speed of 60 miles per hour, and you want to know how long it will take to cover 120 miles. The relationship between distance, rate, and time is given by the equation:

Distance = Rate × Time

Plugging in what we know:

120 = 60t

Solve for t:

t = 120 / 60 = 2

It will take you 2 hours to drive 120 miles. This equation is crucial for planning trips and estimating travel time.

Practice Time!

Now it's your turn. Try solving these equations:

  1. x - 7 = 15

  2. 2x + 3 = 11

  3. (x/5) - 2 = 6

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