Which Of The Following Ordered Pairs Is A Solution Of The Function $f(x)=3x-1$?A) \[$(-1, 2)\$\]B) \[$(-1, -3)\$\]C) \[$(-1, -4)\$\]D) \[$(-1, 4)\$\]
Introduction to Linear Functions
A linear function is a type of mathematical function that represents a straight line on a graph. It is defined as a function of the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will focus on solving for ordered pairs in a linear function, specifically the function f(x) = 3x - 1.
Understanding Ordered Pairs
An ordered pair is a pair of numbers that are used to represent a point on a coordinate plane. It is written in the form (x, y), where x is the x-coordinate and y is the y-coordinate. In the context of a linear function, an ordered pair represents a point on the graph of the function.
Solving for Ordered Pairs in f(x) = 3x - 1
To solve for an ordered pair in the function f(x) = 3x - 1, we need to find a point on the graph of the function that satisfies the equation. We can do this by plugging in a value for x and solving for y.
Let's consider the ordered pairs given in the problem:
A) (-1, 2) B) (-1, -3) C) (-1, -4) D) (-1, 4)
We can plug in x = -1 into the function f(x) = 3x - 1 and solve for y:
f(-1) = 3(-1) - 1 f(-1) = -3 - 1 f(-1) = -4
This tells us that when x = -1, y = -4. Therefore, the ordered pair (-1, -4) is a solution of the function f(x) = 3x - 1.
Evaluating the Other Options
Now, let's evaluate the other options to see if they are also solutions of the function.
Option A: (-1, 2) f(-1) = 3(-1) - 1 f(-1) = -3 - 1 f(-1) = -4 This does not match the y-coordinate of the ordered pair (-1, 2), so option A is not a solution.
Option B: (-1, -3) f(-1) = 3(-1) - 1 f(-1) = -3 - 1 f(-1) = -4 This does not match the y-coordinate of the ordered pair (-1, -3), so option B is not a solution.
Option D: (-1, 4) f(-1) = 3(-1) - 1 f(-1) = -3 - 1 f(-1) = -4 This does not match the y-coordinate of the ordered pair (-1, 4), so option D is not a solution.
Conclusion
In conclusion, the ordered pair (-1, -4) is a solution of the function f(x) = 3x - 1. This is because when x = -1, y = -4, which satisfies the equation of the function.
Tips for Solving for Ordered Pairs
Here are some tips for solving for ordered pairs in a linear function:
- Plug in a value for x: Choose a value for x and plug it into the function to solve for y.
- Solve for y: Use the equation of the function to solve for y.
- Check the y-coordinate: Check that the y-coordinate of the ordered pair matches the value of y that you solved for.
- Evaluate the other options: Evaluate the other options to see if they are also solutions of the function.
Real-World Applications
Solving for ordered pairs in a linear function has many real-world applications. For example:
- Graphing functions: Solving for ordered pairs is an important step in graphing functions. By finding the points on the graph of a function, you can create a visual representation of the function.
- Modeling real-world situations: Linear functions can be used to model real-world situations, such as the cost of producing a product or the distance traveled by an object.
- Solving systems of equations: Solving for ordered pairs is also an important step in solving systems of equations. By finding the points of intersection between two or more functions, you can solve for the values of the variables.
Common Mistakes
Here are some common mistakes to avoid when solving for ordered pairs in a linear function:
- Not plugging in a value for x: Make sure to choose a value for x and plug it into the function to solve for y.
- Not solving for y: Make sure to use the equation of the function to solve for y.
- Not checking the y-coordinate: Make sure to check that the y-coordinate of the ordered pair matches the value of y that you solved for.
- Not evaluating the other options: Make sure to evaluate the other options to see if they are also solutions of the function.
Conclusion
In conclusion, solving for ordered pairs in a linear function is an important step in graphing functions, modeling real-world situations, and solving systems of equations. By following the tips and avoiding the common mistakes outlined in this article, you can become proficient in solving for ordered pairs in a linear function.
Q: What is a linear function?
A: A linear function is a type of mathematical function that represents a straight line on a graph. It is defined as a function of the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
Q: What is an ordered pair?
A: An ordered pair is a pair of numbers that are used to represent a point on a coordinate plane. It is written in the form (x, y), where x is the x-coordinate and y is the y-coordinate.
Q: How do I solve for an ordered pair in a linear function?
A: To solve for an ordered pair in a linear function, you need to find a point on the graph of the function that satisfies the equation. You can do this by plugging in a value for x and solving for y.
Q: What is the equation of a linear function?
A: The equation of a linear function is f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
Q: How do I find the slope of a linear function?
A: To find the slope of a linear function, you need to know the coordinates of two points on the graph of the function. You can use the formula m = (y2 - y1) / (x2 - x1) to find the slope.
Q: How do I find the y-intercept of a linear function?
A: To find the y-intercept of a linear function, you need to know the coordinates of a point on the graph of the function. You can use the equation f(x) = mx + b to find the y-intercept.
Q: What is the difference between a linear function and a quadratic function?
A: A linear function is a type of mathematical function that represents a straight line on a graph, while a quadratic function is a type of mathematical function that represents a parabola on a graph.
Q: How do I graph a linear function?
A: To graph a linear function, you need to find the x and y intercepts of the function. You can use the equation f(x) = mx + b to find the x and y intercepts.
Q: What is the significance of the x and y intercepts of a linear function?
A: The x and y intercepts of a linear function are important because they represent the points on the graph of the function where the line intersects the x and y axes.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to find the values of the variables that satisfy both equations. You can use the method of substitution or the method of elimination to solve a system of linear equations.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations is a set of equations in which the variables are raised to the power of 1, while a system of nonlinear equations is a set of equations in which the variables are raised to a power other than 1.
Q: How do I determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
A: To determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions, you need to examine the equations and determine if they are consistent or inconsistent.
Q: What is the significance of the solution to a system of linear equations?
A: The solution to a system of linear equations is important because it represents the values of the variables that satisfy both equations.
Q: How do I use a system of linear equations to model real-world situations?
A: To use a system of linear equations to model real-world situations, you need to identify the variables and the relationships between them. You can then use the equations to solve for the values of the variables.
Q: What are some common applications of linear functions?
A: Some common applications of linear functions include:
- Cost and revenue analysis: Linear functions can be used to model the cost and revenue of a business.
- Distance and time analysis: Linear functions can be used to model the distance traveled by an object over time.
- Population growth analysis: Linear functions can be used to model the growth of a population over time.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Physics and engineering: Systems of linear equations can be used to model the motion of objects and the forces acting on them.
- Economics: Systems of linear equations can be used to model the relationships between economic variables such as supply and demand.
- Computer science: Systems of linear equations can be used to model the relationships between variables in computer science applications such as data analysis and machine learning.