Solving F(x) = -7 Given F(x) = 3x - 1

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In this article, we will delve into the process of finding the solution set for a given function. Specifically, we will focus on the function F(x) = 3x - 1 and determine the value(s) of x for which F(x) equals -7. This involves applying basic algebraic principles to isolate the variable x and arrive at the solution. Understanding how to solve such equations is fundamental in mathematics and has broad applications in various fields.

To effectively solve for x, it's crucial to understand the nature of the function F(x) = 3x - 1. This is a linear function, characterized by its straight-line graph. The function takes an input x, multiplies it by 3, and then subtracts 1 from the result. This transformation yields the output F(x). Linear functions like this one are ubiquitous in mathematics and are used to model various real-world phenomena, from simple relationships between quantities to more complex systems.

Understanding the components of the function is key to solving the equation. The coefficient 3 represents the slope of the line, indicating how steeply the line rises or falls. The constant term -1 represents the y-intercept, which is the point where the line crosses the y-axis. These elements together define the behavior of the function and influence its values for different inputs.

Our objective is to find the value(s) of x that satisfy the equation F(x) = -7. This means we need to find the x for which the function F(x) evaluates to -7. To do this, we substitute the expression for F(x) into the equation, resulting in the equation 3x - 1 = -7. This equation represents a specific condition that x must meet, and our task is to isolate x and determine its value.

The equation 3x - 1 = -7 is a linear equation, and solving it involves applying inverse operations to both sides of the equation to isolate x. We'll start by adding 1 to both sides to undo the subtraction, and then we'll divide both sides by 3 to undo the multiplication. This process will lead us to the solution for x.

Now, let's proceed with solving the equation 3x - 1 = -7 step by step. The goal is to isolate x on one side of the equation. The first step involves undoing the subtraction of 1 by adding 1 to both sides of the equation. This maintains the equality and moves us closer to isolating x.

Adding 1 to both sides, we get:

3x - 1 + 1 = -7 + 1

Simplifying, we have:

3x = -6

Next, we need to undo the multiplication of 3 by x. To do this, we divide both sides of the equation by 3. This will isolate x and give us its value.

Dividing both sides by 3, we get:

3x / 3 = -6 / 3

Simplifying, we find:

x = -2

Therefore, the solution to the equation 3x - 1 = -7 is x = -2. This means that when x is -2, the function F(x) evaluates to -7.

To ensure the accuracy of our solution, it's crucial to verify that x = -2 indeed satisfies the equation F(x) = -7. This can be done by substituting x = -2 back into the original function and checking if the result is -7. This step is essential in mathematics to confirm the correctness of the solution and catch any potential errors.

Substituting x = -2 into F(x) = 3x - 1, we get:

F(-2) = 3(-2) - 1

Simplifying, we have:

F(-2) = -6 - 1

F(-2) = -7

As we can see, when x = -2, F(x) equals -7, confirming that our solution is correct. This verification step is a good practice in problem-solving, as it provides confidence in the answer and ensures that the solution is accurate.

The solution set is the set of all values of x that satisfy the equation F(x) = -7. In this case, we found that x = -2 is the only solution. Therefore, the solution set is {-2}. This set contains a single element, indicating that there is only one value of x that makes the equation true.

In this article, we successfully found the solution set for the equation F(x) = -7, where F(x) = 3x - 1. By setting up the equation, solving for x, and verifying the solution, we determined that the solution set is {-2}. This process demonstrates the fundamental principles of algebra and the importance of verifying solutions to ensure accuracy. Understanding how to solve such equations is crucial in mathematics and its applications in various fields.

  • Solving for x: The core concept of finding the value of x that satisfies the equation.
  • Function F(x) = 3x - 1: The specific function used in the problem.
  • Solution set: The set of all values of x that satisfy the equation.
  • Linear equation: The type of equation we are dealing with.
  • Algebraic principles: The mathematical rules and operations used to solve the equation.
  • Verifying the solution: The process of checking the accuracy of the solution.
  • Isolating x: The process of getting x by itself on one side of the equation.

Q: What is a function?

A function is a mathematical relationship that assigns each input value to a unique output value. In this case, F(x) = 3x - 1 is a linear function that takes x as input and produces 3x - 1 as output.

Q: How do you solve a linear equation?

A linear equation is solved by isolating the variable on one side of the equation. This is done by applying inverse operations to both sides of the equation until the variable is by itself.

Q: Why is it important to verify the solution?

Verifying the solution ensures that the answer is correct and satisfies the original equation. It helps catch any errors made during the solving process.