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Understanding the Problem

When given a table of values, we can often find a linear equation that represents the relationship between the variables. In this case, we are given a table with two columns, x and y, and we need to find the linear equation that gives the rule for this table.

Analyzing the Table

Let's take a closer look at the table and try to identify any patterns or relationships between the values.

x y
2 6
3 9
4 12
5 15

We can see that as x increases by 1, y also increases by 3. This suggests that the relationship between x and y is linear.

Finding the Slope

To find the linear equation, we need to find the slope of the line. The slope is the change in y divided by the change in x. In this case, the change in y is 3 and the change in x is 1, so the slope is 3/1 = 3.

Finding the y-Intercept

Now that we have the slope, we can use one of the points from the table to find the y-intercept. Let's use the point (2, 6). We can plug this point into the equation y = mx + b, where m is the slope and b is the y-intercept.

6 = 3(2) + b

Simplifying the equation, we get:

6 = 6 + b

Subtracting 6 from both sides, we get:

0 = b

So the y-intercept is 0.

Writing the Linear Equation

Now that we have the slope and the y-intercept, we can write the linear equation. The equation is:

y = 3x

This equation represents the relationship between x and y in the table.

Verifying the Equation

To verify that the equation is correct, we can plug in some of the points from the table and check if they satisfy the equation.

Let's plug in the point (2, 6):

6 = 3(2) 6 = 6

This is true, so the equation is correct.

Let's plug in the point (3, 9):

9 = 3(3) 9 = 9

This is also true, so the equation is correct.

Conclusion

In this article, we found the linear equation that represents the relationship between x and y in the given table. We analyzed the table, found the slope and the y-intercept, and wrote the linear equation. We also verified that the equation is correct by plugging in some of the points from the table.

Linear Equation Representation of a Table: Key Takeaways

  • The linear equation that represents the relationship between x and y in the table is y = 3x.
  • The slope of the line is 3.
  • The y-intercept is 0.
  • The equation can be verified by plugging in some of the points from the table.

Real-World Applications

Linear equations have many real-world applications, such as:

  • Modeling population growth
  • Describing the motion of objects
  • Calculating the cost of goods
  • Determining the amount of interest on a loan

Tips and Tricks

  • When analyzing a table, look for patterns or relationships between the values.
  • Use the slope and the y-intercept to write the linear equation.
  • Verify the equation by plugging in some of the points from the table.

Common Mistakes

  • Failing to identify the slope and the y-intercept.
  • Writing the equation incorrectly.
  • Failing to verify the equation.

Conclusion

In conclusion, linear equations are an important concept in mathematics that have many real-world applications. By understanding how to find the slope and the y-intercept, and writing the linear equation, we can model and analyze many different types of data.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about linear equation representation of a table.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

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

A: To find the slope of a linear equation, you need to find the change in y divided by the change in x. This can be done by using two points from the table and plugging them into the equation (y2 - y1) / (x2 - x1).

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

A: To find the y-intercept of a linear equation, you need to use one of the points from the table and plug it into the equation y = mx + b. Then, solve for b.

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

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: Can I use a linear equation to model a non-linear relationship?

A: No, a linear equation can only model a linear relationship. If the relationship is non-linear, you will need to use a different type of equation, such as a quadratic equation.

Q: How do I verify a linear equation?

A: To verify a linear equation, you need to plug in some of the points from the table and check if they satisfy the equation.

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

A: Some real-world applications of linear equations include:

  • Modeling population growth
  • Describing the motion of objects
  • Calculating the cost of goods
  • Determining the amount of interest on a loan

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

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

  • Failing to identify the slope and the y-intercept
  • Writing the equation incorrectly
  • Failing to verify the equation

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to use a coordinate plane and plot the points that satisfy the equation. Then, draw a line through the points to represent the equation.

Q: Can I use a linear equation to model a relationship between two variables?

A: Yes, a linear equation can be used to model a relationship between two variables. However, you will need to make sure that the relationship is linear.

Q: How do I determine if a relationship is linear or non-linear?

A: To determine if a relationship is linear or non-linear, you need to examine the data and look for patterns or trends. If the relationship is linear, you can use a linear equation to model it. If the relationship is non-linear, you will need to use a different type of equation.

Conclusion

In conclusion, linear equations are an important concept in mathematics that have many real-world applications. By understanding how to find the slope and the y-intercept, and writing the linear equation, we can model and analyze many different types of data.