Solving For X In Linear Equations And Cuboid Volume Calculation

In this section, we will meticulously walk through the process of solving the linear equation $2x - 3 = 11$ to determine the value of $x$. Linear equations are fundamental in algebra, and mastering the techniques to solve them is crucial for more advanced mathematical concepts. Our goal is to isolate the variable $x$ on one side of the equation, revealing its numerical value. We will achieve this by applying a series of algebraic operations, ensuring that each step maintains the balance and equality of the equation. This detailed explanation aims to provide a comprehensive understanding of the step-by-step solution, making it clear and accessible for learners of all levels. The process involves adding the same value to both sides, simplifying, and then dividing to finally isolate $x$. By the end of this section, you will have a solid grasp of how to solve similar equations with confidence and precision.

Step 1: Isolate the Term with $x$

To isolate the term containing $x$, which is $2x$, we need to eliminate the constant term, $-3$, from the left side of the equation. We achieve this by performing the inverse operation: adding $3$ to both sides of the equation. This maintains the equation's balance, as whatever operation we apply to one side, we must apply to the other.

So, we start with the equation:

$$2x - 3 = 11$$

Adding $3$ to both sides, we get:

$$2x - 3 + 3 = 11 + 3$$

This simplifies to:

$$2x = 14$$

Step 2: Solve for $x$

Now that we have isolated the term $2x$, we need to isolate $x$ itself. Currently, $x$ is being multiplied by $2$. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by $2$ to maintain balance.

Starting with:

$$2x = 14$$

Divide both sides by $2$:

$$\frac{2x}{2} = \frac{14}{2}$$

This simplifies to:

$$x = 7$$

Therefore, the value of $x$ that satisfies the equation $2x - 3 = 11$ is $7$.

Verification

To ensure our solution is correct, we substitute $x = 7$ back into the original equation and check if it holds true.

Original equation:

$$2x - 3 = 11$$

Substitute $x = 7$:

$$2(7) - 3 = 11$$

$$14 - 3 = 11$$

$$11 = 11$$

Since the equation holds true, our solution $x = 7$ is correct. This verification step is crucial in mathematics to confirm the accuracy of our calculations and ensure that the value we found indeed satisfies the given conditions. By substituting the solution back into the original equation, we provide a clear and undeniable confirmation of its correctness.

This section focuses on calculating the volume of a cuboid, a fundamental concept in three-dimensional geometry. A cuboid, also known as a rectangular prism, is a three-dimensional shape with six rectangular faces. Understanding how to calculate its volume is essential in various fields, including engineering, architecture, and everyday problem-solving. The volume of a cuboid represents the amount of space it occupies and is measured in cubic units. To find the volume, we use a simple formula that involves the cuboid's length, width, and height. This section will guide you through the process step-by-step, providing a clear and concise explanation that will enable you to calculate the volume of any cuboid, given its dimensions. By the end of this section, you will have a solid understanding of how to apply the formula and interpret the result in cubic centimeters ($cm^3$). This knowledge is not only valuable for academic purposes but also for practical applications in real-world scenarios.

Understanding the Formula

The volume of a cuboid is calculated using the formula:

$$Volume = Length \times Width \times Height$$

Where:

• Length is the distance from one end of the cuboid to the other.
• Width is the distance from one side of the cuboid to the other.
• Height is the vertical distance of the cuboid.

All these dimensions must be in the same unit for the calculation to be accurate. In this case, the dimensions are given in centimeters (cm), so the volume will be in cubic centimeters ($cm^3$).

Applying the Formula

Given the dimensions of the cuboid:

• Length = 5 cm
• Width = 4 cm
• Height = 3 cm

We can substitute these values into the formula:

$$Volume = 5 \text{ cm} \times 4 \text{ cm} \times 3 \text{ cm}$$

Calculation

Multiply the length, width, and height together:

$$Volume = 5 \times 4 \times 3 \text{ cm}^3$$

$$Volume = 20 \times 3 \text{ cm}^3$$

$$Volume = 60 \text{ cm}^3$$

Therefore, the volume of the cuboid is 60 cubic centimeters ($cm^3$). This calculation demonstrates a straightforward application of the volume formula, highlighting how multiplying the three dimensions provides the total space enclosed within the cuboid. Understanding this concept is crucial for visualizing and quantifying three-dimensional space, which is a fundamental aspect of geometry and its applications in various practical fields.

Conclusion

In conclusion, we have successfully found the value of $x$ in the equation $2x - 3 = 11$, which is $x = 7$. We also calculated the volume of a cuboid with dimensions 5 cm long, 4 cm wide, and 3 cm high, which is 60 $cm^3$. These exercises demonstrate the application of fundamental algebraic and geometric principles, providing a solid foundation for tackling more complex problems in mathematics and related fields. Mastering these basic calculations is essential for developing a strong mathematical intuition and problem-solving skills.