Introduction
Hey guys! Today, we're diving into some math problems that involve calculating costs and simplifying expressions. We'll tackle a problem about fencing a flower garden and another one about simplifying an algebraic expression. Let's get started!
Calculating the Cost to Fence a Flower Garden
In this section, we'll explore how to calculate the cost of fencing a flower garden using a given formula. The formula helps us determine the total cost based on a variable, which in this case, represents a specific measurement related to the garden. Understanding how to use such formulas is crucial for real-world applications, whether you're planning a home improvement project or managing a larger landscaping endeavor. Let's break down the problem step by step to make sure we fully grasp the concept.
Understanding the Formula
The problem states that the cost C (in Rands) to fence the flower garden is given by the formula C = 15x + 10. In this formula:
- C represents the total cost of fencing the garden.
- x is a variable, which likely represents a measurement related to the garden, such as the length of the fence needed.
- 15 is the cost per unit of x (e.g., cost per meter of fencing).
- 10 is a fixed cost, which could represent a base fee for materials or labor.
This formula is a linear equation, which means that the cost C increases linearly with the value of x. The slope of the line is 15, indicating the rate at which the cost increases for each unit increase in x. The y-intercept is 10, representing the fixed cost when x is 0.
Applying the Formula with x = 3
The problem specifies that x = 3. To calculate the cost, we need to substitute this value into the formula:
C = 15x + 10
Substitute x with 3:
C = 15(3) + 10
Now, we perform the multiplication:
C = 45 + 10
Finally, we add the numbers together:
C = 55
So, the cost to fence the flower garden when x = 3 is 55 Rands.
Interpreting the Result
Our calculation shows that it will cost 55 Rands to fence the garden when x is 3. This result is crucial for budgeting and planning purposes. If x represents the length in meters, then we know that each meter of fencing costs 15 Rands, and there is an additional fixed cost of 10 Rands. Understanding these components helps in making informed decisions about the project. For example, if you need to increase the size of the garden, you can easily estimate the additional cost by multiplying the extra length by 15 and adding it to the current cost.
Real-World Applications
The ability to use and interpret formulas like this is incredibly useful in many real-world scenarios. Whether you're calculating the cost of materials for a home renovation, estimating the budget for a landscaping project, or determining the expenses for a business venture, these skills are essential. By breaking down the formula and understanding each component, you can accurately predict costs and manage your resources effectively. Furthermore, understanding linear equations helps in visualizing how costs change with different variables, enabling better financial planning and decision-making.
Simplifying Algebraic Expressions
Now, let's switch gears and dive into simplifying algebraic expressions. Simplifying expressions is a fundamental skill in algebra that involves combining like terms and performing operations to reduce an expression to its simplest form. This skill is essential for solving equations, understanding algebraic concepts, and tackling more complex mathematical problems. We'll break down the simplification process step by step, making it easy to follow along and master the technique.
Understanding the Expression
The expression we need to simplify is: 4(x + 2) + 3(2x - 1). This expression involves variables, constants, and operations such as multiplication and addition. To simplify it, we need to apply the distributive property and combine like terms.
Applying the Distributive Property
The distributive property states that a(b + c) = ab + ac. We'll use this property to remove the parentheses in our expression.
First, distribute the 4 across (x + 2):
4(x + 2) = 4 * x + 4 * 2 = 4x + 8
Next, distribute the 3 across (2x - 1):
3(2x - 1) = 3 * 2x + 3 * (-1) = 6x - 3
Now, our expression looks like this:
4x + 8 + 6x - 3
Combining Like Terms
Like terms are terms that have the same variable raised to the same power. In our expression, 4x and 6x are like terms, and 8 and -3 are like terms (constants). We can combine these terms by adding their coefficients.
Combine the x terms:
4x + 6x = (4 + 6)x = 10x
Combine the constants:
8 - 3 = 5
So, the simplified expression is:
10x + 5
Understanding the Simplified Expression
The simplified expression, 10x + 5, is equivalent to the original expression, 4(x + 2) + 3(2x - 1), but it is in a more concise and manageable form. This simplified form makes it easier to substitute values for x, solve equations, and analyze the expression's behavior. For instance, if we want to find the value of the expression when x = 2, we can simply substitute x into the simplified expression:
10(2) + 5 = 20 + 5 = 25
Real-World Applications
Simplifying algebraic expressions is a fundamental skill that has numerous applications in mathematics and various fields. It is used in solving equations, graphing functions, and modeling real-world situations. Whether you're working on a physics problem, designing a structure, or analyzing data, the ability to simplify expressions is crucial for making calculations and drawing conclusions. Furthermore, mastering this skill lays the foundation for more advanced algebraic concepts, such as factoring and solving systems of equations.
Conclusion
So, there you have it! We've calculated the cost to fence a flower garden using a given formula and simplified an algebraic expression. These are essential skills in mathematics that have practical applications in everyday life. Keep practicing, and you'll become a math whiz in no time!
