Solve For X And Area Calculations: Math Problems

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Hey guys! Let's dive into some math problems today. We're going to tackle solving for 'x' in a few different equations and then calculate the area of some shapes. Ready? Let's get started!

Solving for X in Equations

Alright, so the first part involves finding the value of 'x' in a series of equations. The key here is to isolate 'x' on one side of the equation. We'll do this by performing inverse operations. Remember, whatever you do to one side, you gotta do to the other! Let's break down each equation step by step. Keep in mind the importance of algebraic manipulation to simplify and rearrange the equation in order to isolate 'x'. These steps often involve addition, subtraction, multiplication, and division. Always double-check your work by substituting the solved value of 'x' back into the original equation to ensure it holds true. Understanding these principles is crucial for success in algebra and beyond. So let's begin with a detailed breakdown of each equation:

  1. x + 7 = 20

    To find 'x', we need to get rid of that '+ 7'. So, we subtract 7 from both sides:

    x + 7 - 7 = 20 - 7

    x = 13

  2. 2x + 12 = 30

    First, let's get rid of the '+ 12' by subtracting 12 from both sides:

    2x + 12 - 12 = 30 - 12

    2x = 18

    Now, we have 2x = 18. To isolate 'x', we divide both sides by 2:

    2x / 2 = 18 / 2

    x = 9

  3. 3x - 32 = 140

    Let's tackle this one! First, add 32 to both sides to get rid of the '- 32':

    3x - 32 + 32 = 140 + 32

    3x = 172

    Now, divide both sides by 3 to solve for 'x':

    3x / 3 = 172 / 3

    x = 57.33 (approximately)

  4. 2x - 12 = 214

    Add 12 to both sides to isolate the term with 'x':

    2x - 12 + 12 = 214 + 12

    2x = 226

    Divide both sides by 2:

    2x / 2 = 226 / 2

    x = 113

  5. 6 - x - 30 = 870

    Combine the constants on the left side:

    -x - 24 = 870

    Add 24 to both sides:

    -x = 894

    Multiply both sides by -1 to solve for 'x':

    x = -894

Calculating Areas of Figures

Now, let's switch gears and calculate the area of some shapes. Remember, the area is the amount of space inside a 2D shape. We'll need to use the appropriate formulas for each shape.

a) Figure A

This figure appears to be a square or a rectangle with one side labeled as 10m. Assuming it's a square (since only one side is given):

  • Area of a square = side * side
  • Area = 10m * 10m
  • Area = 100 m²

So, the area of figure A is 100 square meters. When calculating the area, it's important to identify the shape accurately. For rectangles and squares, the formula is straightforward: length multiplied by width. Triangles require multiplying the base by the height and then dividing by two. Circles involve using the value of pi (approximately 3.14159) in the formula πr², where 'r' is the radius of the circle. Understanding the specific properties of each shape is crucial for selecting the correct formula and obtaining accurate area measurements. Always remember to include the appropriate unit of measurement, such as square meters (m²) or square feet (ft²), to provide a complete and meaningful answer.

b) Figure B

This figure looks like a combination of rectangles. We have a rectangle with sides 4m and 8m, and another rectangle with one side 5m (we'll assume the other side is 4m based on the diagram). Let's find the area of each rectangle and then add them together.

  • Area of Rectangle 1 = length * width = 4m * 8m = 32 m²
  • Area of Rectangle 2 = length * width = 4m * 5m = 20 m²
  • Total Area = Area of Rectangle 1 + Area of Rectangle 2 = 32 m² + 20 m² = 52 m²

The total area of figure B is 52 square meters. Breaking down complex shapes into simpler components, like rectangles or triangles, is a common strategy for calculating area. Once the individual areas are found, summing them up gives the total area of the composite shape. This approach simplifies the problem and allows for easier computation. Additionally, be mindful of units when adding areas; they must be consistent (e.g., all in square meters) to obtain a meaningful result. Such decomposition techniques are widely applicable in various fields, including architecture, engineering, and design, where complex shapes are frequently encountered. Always double-check measurements and calculations to ensure accuracy, especially when dealing with real-world applications.

Conclusion

And there you have it! We've solved for 'x' in several equations and calculated the areas of two figures. Remember, practice makes perfect, so keep working on these types of problems to sharpen your skills. Whether it's isolating variables in equations or applying area formulas, a solid understanding of these concepts is super useful. And remember, it’s important to double-check your work to avoid simple mistakes. Keep up the great work, and you'll be a math whiz in no time! If you guys have any questions, feel free to ask. Happy calculating!