Math Examples Explained: Your Quick Guide!

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Hey there, math whizzes! Ready to tackle some examples? Let's dive into some cool mathematical problems with easy-to-understand explanations. This is your go-to guide for getting those problems solved and understanding the 'why' behind the 'what.' We'll break down each example step-by-step, making sure you feel confident about every concept. No confusing jargon, just clear, concise explanations to boost your understanding. Get ready to flex those math muscles and ace those quizzes, guys! Let's get started with our first example – it's going to be a blast!

Example 1: Solving a Simple Algebraic Equation

Alright, let's start with a classic: solving an algebraic equation. This is a fundamental skill in math, and once you get the hang of it, you'll be solving equations like a pro! Our example is a pretty straightforward one, to ease you in. We will use the equation: 3x + 5 = 14. Our goal is to find the value of 'x' that makes this equation true. Think of it like this: we're trying to find the secret number that, when multiplied by 3 and then added to 5, gives us 14. Sounds like a fun puzzle, right?

First things first, to solve for 'x', we need to isolate it. This means getting 'x' by itself on one side of the equation. We start by getting rid of the '+ 5.' To do this, we subtract 5 from both sides of the equation. Why both sides? Because in algebra, whatever you do to one side of the equation, you must do to the other to keep it balanced. It's like a seesaw – if you only add or remove weight from one side, it will tip! So, our equation becomes: 3x + 5 - 5 = 14 - 5. This simplifies to 3x = 9.

Now, we have 3x = 9. This means 3 times 'x' equals 9. To find the value of 'x,' we do the opposite of multiplication, which is division. We divide both sides of the equation by 3. This gives us: 3x / 3 = 9 / 3. Simplifying, we get x = 3. Ta-da! We've solved the equation. The value of 'x' that makes the original equation true is 3. To check our work, we can substitute 'x' with 3 in the original equation: 3 * 3 + 5 = 9 + 5 = 14. And that’s what we wanted!

So, in this example, we’ve covered the fundamentals. Remember to always keep the equation balanced and perform the opposite operation to isolate the variable. Once you master this process, you will be well on your way to understanding more complex algebra.

Step-by-Step Breakdown:

  1. Original Equation: 3x + 5 = 14
  2. Subtract 5 from both sides: 3x = 9
  3. Divide both sides by 3: x = 3
  4. Solution: x = 3

Example 2: Calculating the Area of a Triangle

Let’s switch gears and look at geometry! Calculating the area of a triangle is a super useful skill. Whether you're a budding architect, a budding designer, or just love solving problems, knowing how to find the area of a triangle comes in handy. Remember the formula, and you're golden!

The formula for the area of a triangle is remarkably simple: Area = 0.5 * base * height, or more commonly written as Area = (1/2) * base * height. The 'base' is the length of the bottom side of the triangle, and the 'height' is the perpendicular distance from the base to the opposite vertex (the highest point). Imagine you have a triangle with a base of 10 cm and a height of 6 cm. Let's find its area.

We start by plugging the values into our formula. The base is 10 cm, and the height is 6 cm. So, the formula looks like this: Area = (1/2) * 10 cm * 6 cm. First, we multiply 10 cm by 6 cm, which equals 60 square centimeters. Then, we multiply that by 1/2 (or divide by 2). This gives us Area = (1/2) * 60 square centimeters = 30 square centimeters. So, the area of the triangle is 30 square centimeters. Remember, area is always measured in square units, because we are essentially calculating the space a two-dimensional shape takes up. This is a very important fact to always keep in mind.

If the triangle isn't a right-angled triangle, make sure to measure the perpendicular height, not the length of a slanted side. The perpendicular height is the shortest distance from the base to the top vertex, forming a 90-degree angle with the base. Also, it’s worth noting that this formula works for any type of triangle – whether it's equilateral, isosceles, or scalene. It's a versatile tool to have in your mathematical toolkit! Practice a few different triangles with different base and height measurements, and you'll get the hang of it quickly.

Step-by-Step Breakdown:

  1. Identify Base and Height: Base = 10 cm, Height = 6 cm
  2. Apply Formula: Area = (1/2) * base * height
  3. Calculate: Area = (1/2) * 10 cm * 6 cm = 30 sq cm
  4. Solution: Area = 30 sq cm

Example 3: Finding the Percentage of a Number

Let's wrap things up with a practical example: finding the percentage of a number. Percentages are everywhere in the real world – from sales discounts to interest rates. So, understanding how to calculate them is super important. We’ll break this down so it’s easy peasy.

Suppose you want to find 20% of 80. This is a classic type of problem, frequently used in daily life. There are a few ways to approach this. The most straightforward way is to convert the percentage to a decimal and then multiply it by the number. To convert a percentage to a decimal, you divide it by 100. So, 20% becomes 20 / 100 = 0.20. Now, you multiply this decimal by the number you're finding the percentage of. In this case, that number is 80. So, the calculation looks like this: 0.20 * 80.

When you multiply 0.20 by 80, you get 16. That means 20% of 80 is 16. Another way to think about this is to set up a proportion: (percentage / 100) = (part / whole). In our example, (20 / 100) = (x / 80). Then, you cross-multiply: 20 * 80 = 100 * x. This gives you 1600 = 100x. To solve for x, you divide both sides by 100, which gives you x = 16. So, regardless of the method, you end up with the same answer.

Percentages are essential for understanding data, comparing values, and making informed decisions. Mastering this skill gives you a leg up in everyday situations, from shopping for the best deals to understanding financial reports. Practice with different percentages and numbers, and soon it will become second nature! You will be well on your way to mastering these concepts!

Step-by-Step Breakdown:

  1. Convert Percentage to Decimal: 20% = 0.20
  2. Multiply Decimal by the Number: 0.20 * 80
  3. Calculate: 0.20 * 80 = 16
  4. Solution: 20% of 80 is 16

I hope that was helpful, folks! Remember, practice makes perfect. Keep exploring, keep questioning, and you'll become math masters in no time! Let me know if you want more examples or have any questions. Happy calculating!