Given F(x)=5(2-x), What Is The Value Of X When F(x) = 25

Mar 11, 2025 by ADMIN 57 views

**Solving for x in a Linear Function: A Step-by-Step Guide**

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Introduction

In mathematics, functions are used to describe the relationship between variables. A linear function is a type of function that can be represented by a straight line on a graph. In this article, we will explore how to solve for x in a linear function, using the given function f(x) = 5(2 - x) as an example.

Understanding the Function

The given function is f(x) = 5(2 - x). This function can be rewritten as f(x) = 10 - 5x. To understand the function, let's break it down into its components:

The function has a constant term of 10, which means that when x is 0, the function will evaluate to 10.
The function has a coefficient of -5, which means that for every unit increase in x, the function will decrease by 5 units.

Solving for x

To solve for x, we need to isolate x on one side of the equation. In this case, we are given that f(x) = 25. We can substitute this value into the function and solve for x.

f(x) = 25 10 - 5x = 25

Isolating x

To isolate x, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting 10 from both sides of the equation.

10 - 5x - 10 = 25 - 10 -5x = 15

Solving for x

Now that we have isolated x, we can solve for its value. We can do this by dividing both sides of the equation by -5.

-5x / -5 = 15 / -5 x = -3

Conclusion

In this article, we have explored how to solve for x in a linear function using the given function f(x) = 5(2 - x) as an example. We have broken down the function into its components, isolated x, and solved for its value. The final answer is x = -3.

Example Use Cases

Solving for x in a linear function has many practical applications in real-world scenarios. Here are a few examples:

Physics: In physics, linear functions are used to describe the motion of objects. Solving for x can help us determine the position of an object at a given time.
Economics: In economics, linear functions are used to describe the relationship between variables such as price and quantity demanded. Solving for x can help us determine the optimal price and quantity for a product.
Computer Science: In computer science, linear functions are used to describe the relationship between variables such as time and space complexity. Solving for x can help us determine the optimal algorithm for a problem.

Tips and Tricks

Here are a few tips and tricks to help you solve for x in a linear function:

Use algebraic manipulation: Algebraic manipulation is a powerful tool for solving for x. Use it to isolate x and solve for its value.
Check your work: Always check your work to make sure that you have solved for x correctly.
Use a calculator: If you are having trouble solving for x, try using a calculator to check your work.

Conclusion

In conclusion, solving for x in a linear function is a fundamental concept in mathematics. By understanding the function, isolating x, and solving for its value, we can apply this concept to a wide range of real-world scenarios. Remember to use algebraic manipulation, check your work, and use a calculator if needed to ensure that you have solved for x correctly.

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Q: What is a linear function?

A: A linear function is a type of function that can be represented by a straight line on a graph. It is a function that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Q: How do I know if a function is linear?

A: To determine if a function is linear, look for the following characteristics:

The function can be written in the form f(x) = mx + b.
The graph of the function is a straight line.
The function has a constant rate of change.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function that can be represented by a straight line on a graph, while a quadratic function is a function that can be represented by a parabola on a graph. Quadratic functions have a variable rate of change, whereas linear functions have a constant rate of change.

Q: How do I solve for x in a linear function?

A: To solve for x in a linear function, follow these steps:

Write the function in the form f(x) = mx + b.
Isolate x by subtracting b from both sides of the equation.
Divide both sides of the equation by m.

Q: What if the function is in the form f(x) = a(x - h)^2 + k?

A: If the function is in the form f(x) = a(x - h)^2 + k, it is a quadratic function, not a linear function. To solve for x, you will need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: Can I use a calculator to solve for x?

A: Yes, you can use a calculator to solve for x. In fact, calculators are often the fastest and most accurate way to solve for x. However, make sure to check your work to ensure that the calculator is giving you the correct answer.

Q: What if I get a negative value for x?

A: If you get a negative value for x, it means that the function is not defined for that value of x. This can happen if the function has a domain restriction, such as x > 0.

Q: Can I use algebraic manipulation to solve for x?

A: Yes, you can use algebraic manipulation to solve for x. In fact, algebraic manipulation is often the best way to solve for x, as it allows you to isolate x and solve for its value.

Q: What if I get a complex value for x?

A: If you get a complex value for x, it means that the function is not defined for that value of x. This can happen if the function has a complex root, such as x = a + bi.