The Equation of the Second Table

Oct 13, 2025 by talynt.org 97 views

Introduction

Linear functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one. In this article, we will explore two linear functions represented by tables and discuss how to determine the equation of the second table.

The First Table: A Given Linear Equation

The first table represents a linear equation, which is given as:

${ y = 5.75x + 34.5 }$

This equation represents a straight line with a slope of 5.75 and a y-intercept of 34.5. The slope-intercept form of a linear equation is given by:

${ y = mx + b }$

where m is the slope and b is the y-intercept.

The Second Table: An Unknown Linear Equation

The second table represents an unknown linear equation. We are asked to determine the equation of this table.

x	y
2	12
4	23
6	34

To determine the equation of the second table, we need to find the slope and y-intercept of the line that passes through the given points.

Finding the Slope

The slope of a line can be found using the formula:

${ m = \frac{y_2 - y_1}{x_2 - x_1} }$

where (x1, y1) and (x2, y2) are two points on the line.

Using the given points (2, 12) and (4, 23), we can find the slope as follows:

${ m = \frac{23 - 12}{4 - 2} = \frac{11}{2} = 5.5 }$

Finding the Y-Intercept

The y-intercept of a line can be found using the formula:

${ b = y - mx }$

where (x, y) is a point on the line and m is the slope.

Using the given point (2, 12) and the slope (5.5), we can find the y-intercept as follows:

${ b = 12 - 5.5(2) = 12 - 11 = 1 }$

The Equation of the Second Table

Now that we have found the slope and y-intercept of the line that passes through the given points, we can write the equation of the second table as:

${ y = 5.5x + 1 }$

Conclusion

In this article, we have discussed two linear functions represented by tables. We have determined the equation of the first table, which is given as y = 5.75x + 34.5. We have also determined the equation of the second table, which is y = 5.5x + 1. The process of finding the equation of a linear function involves finding the slope and y-intercept of the line that passes through the given points.

What is the Value of the Linear Equation Represented by the Second Table?

The value of the linear equation represented by the second table is y = 5.5x + 1.

What is the Slope of the Linear Equation Represented by the Second Table?

The slope of the linear equation represented by the second table is 5.5.

What is the Y-Intercept of the Linear Equation Represented by the Second Table?

The y-intercept of the linear equation represented by the second table is 1.

What is the Equation of the Linear Function Represented by the First Table?

The equation of the linear function represented by the first table is y = 5.75x + 34.5.

What is the Slope of the Linear Function Represented by the First Table?

The slope of the linear function represented by the first table is 5.75.

What is the Y-Intercept of the Linear Function Represented by the First Table?

The y-intercept of the linear function represented by the first table is 34.5.

The Importance of Linear Functions

Linear functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. They are used to model real-world phenomena, such as the cost of producing a product, the demand for a product, and the supply of a product.

The Applications of Linear Functions

Linear functions have numerous applications in various fields, including:

Science: Linear functions are used to model the growth of populations, the spread of diseases, and the decay of radioactive materials.
Engineering: Linear functions are used to design and optimize systems, such as electrical circuits, mechanical systems, and communication systems.
Economics: Linear functions are used to model the demand for a product, the supply of a product, and the cost of producing a product.
Computer Science: Linear functions are used to model the time complexity of algorithms, the space complexity of algorithms, and the performance of computer systems.

The Future of Linear Functions

Linear functions will continue to play a crucial role in various fields, including science, engineering, and economics. As technology advances, the use of linear functions will become more widespread, and new applications will emerge.

Conclusion

Q: What is a linear function?

A: A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one. It is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope of a linear function?

A: The slope of a linear function is a measure of how steep the line is. It is calculated as the change in y divided by the change in x. The slope is denoted by the letter m.

Q: How do I find the slope of a linear function?

A: To find the slope of a linear function, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function is the point where the line crosses the y-axis. It is the value of y when x is equal to zero. The y-intercept is denoted by the letter b.

Q: How do I find the y-intercept of a linear function?

A: To find the y-intercept of a linear function, you can use the formula b = y - mx, where (x, y) is a point on the line and m is the slope.

Q: What is the equation of a linear function?

A: The equation of a linear function is a mathematical expression that describes the relationship between the variables x and y. It is typically written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I write the equation of a linear function?

A: To write the equation of a linear function, you need to know the slope and y-intercept of the line. You can use the slope-intercept form of a linear equation, which is y = mx + b.

Q: What are some examples of linear functions?

A: Some examples of linear functions include:

y = 2x + 3
y = 5x - 2
y = x + 1

Q: What are some real-world applications of linear functions?

A: Linear functions have many real-world applications, including:

Modeling the growth of populations
Describing the cost of producing a product
Calculating the demand for a product
Determining the supply of a product
Modeling the spread of diseases
Calculating the decay of radioactive materials

Q: How do I graph a linear function?

A: To graph a linear function, you can use a coordinate plane and plot the points that satisfy the equation. You can also use a graphing calculator or a computer program to graph the function.

Q: What are some common mistakes to avoid when working with linear functions?

A: Some common mistakes to avoid when working with linear functions include:

Not checking the units of the variables
Not checking the signs of the coefficients
Not checking the domain and range of the function
Not checking for extraneous solutions

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the following methods:

Substitution method
Elimination method
Graphing method
Matrix method

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

A: A linear function is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. A nonlinear function is a function that cannot be written in this form. Nonlinear functions can be quadratic, cubic, or higher-degree functions.

Q: How do I determine if a function is linear or nonlinear?

A: To determine if a function is linear or nonlinear, you can check if it can be written in the form y = mx + b. If it can, then it is a linear function. If it cannot, then it is a nonlinear function.