Fill In The Table Based On The Equation \[$ Y = \frac{1}{3} X + 4 \$\].$\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & $\frac{1}{3} X+4$ & $y$ & $(x, Y)$ \\ \hline & & & \\ \hline & & & \\ \hline & & &

Introduction

In mathematics, equations are used to represent relationships between variables. One common type of equation is a linear equation, which can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this article, we will explore how to fill in a table based on the equation $y = \frac{1}{3} x + 4$. We will also discuss the importance of understanding linear equations and how they are used in real-world applications.

Understanding the Equation

The given equation is $y = \frac{1}{3} x + 4$. This equation represents a linear relationship between the variables $x$ and $y$. The slope of the equation is $\frac{1}{3}$, which means that for every unit increase in $x$, $y$ increases by $\frac{1}{3}$ unit. The y-intercept of the equation is $4$, which means that when $x = 0$, $y = 4$.

Filling in the Table

To fill in the table, we need to substitute different values of $x$ into the equation and calculate the corresponding values of $y$. We will start by filling in the first row of the table.

First Row

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0

To fill in the first row, we substitute $x = 0$ into the equation. This gives us:

$$y = \frac{1}{3} (0) + 4 = 4$$

So, the first row of the table is:

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0	4	4	(0, 4)

Second Row

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0
3

To fill in the second row, we substitute $x = 3$ into the equation. This gives us:

$$y = \frac{1}{3} (3) + 4 = 5$$

So, the second row of the table is:

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0	4	4	(0, 4)
3	5	5	(3, 5)

Third Row

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0
3
6

To fill in the third row, we substitute $x = 6$ into the equation. This gives us:

$$y = \frac{1}{3} (6) + 4 = 6$$

So, the third row of the table is:

$x$	$\frac{1}{3} x+4$	$y$	$(x, y)$
0	4	4	(0, 4)
3	5	5	(3, 5)
6	6	6	(6, 6)

Conclusion

In this article, we filled in a table based on the equation $y = \frac{1}{3} x + 4$. We substituted different values of $x$ into the equation and calculated the corresponding values of $y$. We also discussed the importance of understanding linear equations and how they are used in real-world applications. By filling in the table, we were able to visualize the relationship between the variables $x$ and $y$ and see how the equation represents a linear relationship.

Real-World Applications

Linear equations have many real-world applications. For example, they can be used to model the cost of producing a product, the demand for a product, or the supply of a product. They can also be used to model the relationship between two variables, such as the relationship between the amount of money spent on advertising and the number of sales.

Tips and Tricks

When filling in a table based on a linear equation, it's a good idea to start by filling in the first row and then work your way down. This will help you to see the pattern of the equation and make it easier to fill in the rest of the table.

Practice Problems

Try filling in the table based on the equation $y = 2x - 3$. Use the same format as the table above and fill in the first three rows.

$x$	$2x-3$	$y$	$(x, y)$
0
2
4

Q: What is the equation $y = \frac{1}{3} x + 4$?

A: The equation $y = \frac{1}{3} x + 4$ is a linear equation that represents a relationship between the variables $x$ and $y$. The slope of the equation is $\frac{1}{3}$, which means that for every unit increase in $x$, $y$ increases by $\frac{1}{3}$ unit. The y-intercept of the equation is $4$, which means that when $x = 0$, $y = 4$.

Q: How do I fill in the table based on the equation?

A: To fill in the table, you need to substitute different values of $x$ into the equation and calculate the corresponding values of $y$. You can start by filling in the first row and then work your way down. Make sure to use the same format as the table above and fill in the values of $x$, $\frac{1}{3} x+4$, $y$, and $(x, y)$.

Q: What if I get stuck while filling in the table?

A: If you get stuck while filling in the table, try to identify the pattern of the equation. Look for the slope and y-intercept of the equation and use them to help you fill in the table. You can also try using a calculator or a graphing tool to help you visualize the equation and fill in the table.

Q: Can I use the equation $y = \frac{1}{3} x + 4$ to model real-world situations?

A: Yes, you can use the equation $y = \frac{1}{3} x + 4$ to model real-world situations. For example, you can use it to model the cost of producing a product, the demand for a product, or the supply of a product. You can also use it to model the relationship between two variables, such as the relationship between the amount of money spent on advertising and the number of sales.

Q: How can I use the table to visualize the equation?

A: You can use the table to visualize the equation by looking at the values of $x$ and $y$ that you filled in. You can also use a graphing tool or a calculator to graph the equation and see how it relates to the values in the table.

Q: Can I use the equation $y = \frac{1}{3} x + 4$ to solve problems?

A: Yes, you can use the equation $y = \frac{1}{3} x + 4$ to solve problems. For example, you can use it to solve problems that involve finding the cost of producing a product, the demand for a product, or the supply of a product. You can also use it to solve problems that involve finding the relationship between two variables, such as the relationship between the amount of money spent on advertising and the number of sales.

Q: How can I practice using the equation $y = \frac{1}{3} x + 4$?

A: You can practice using the equation $y = \frac{1}{3} x + 4$ by filling in tables, graphing the equation, and solving problems that involve the equation. You can also try using the equation to model real-world situations and see how it relates to the values in the table.

Q: Can I use the equation $y = \frac{1}{3} x + 4$ to create a linear model?

A: Yes, you can use the equation $y = \frac{1}{3} x + 4$ to create a linear model. A linear model is a mathematical model that uses a linear equation to describe the relationship between two variables. You can use the equation $y = \frac{1}{3} x + 4$ to create a linear model that describes the relationship between the variables $x$ and $y$.

Q: How can I use the equation $y = \frac{1}{3} x + 4$ to make predictions?

A: You can use the equation $y = \frac{1}{3} x + 4$ to make predictions by using the values of $x$ and $y$ that you filled in to make predictions about the relationship between the variables. For example, you can use the equation to predict the cost of producing a product, the demand for a product, or the supply of a product.

Q: Can I use the equation $y = \frac{1}{3} x + 4$ to solve systems of equations?

A: Yes, you can use the equation $y = \frac{1}{3} x + 4$ to solve systems of equations. A system of equations is a set of two or more equations that involve the same variables. You can use the equation $y = \frac{1}{3} x + 4$ to solve systems of equations that involve the variables $x$ and $y$.